Rigid analytic curves and their Jacobians

Rigid analytic curves and theirJacobians

Dissertationzur Erlangung des Doktorgrades Dr. rer. nat. der Fakultat

fur Mathematik und Wirtschaftswissenschaften derUniversitat Ulm

vorgelegt vonSophie Schmieg

ausEbersberg

Ulm 2013

Erstgutachter:Prof. Dr. Werner LutkebohmertZweitgutachter:Prof. Dr. Stefan Wewers

Amtierender Dekan:Prof. Dr. Dieter Rautenbach

Tag der Promotion: 19. Juni 2013

Contents

Glossary of Notations vii

Introduction ix1. The Jacobian of a curve in the complex case . . . . . . . . . . . . ix2. Mumford curves and general rigid analytic curves . . . . . . . . . ix3. Outline of the chapters and the results of this work . . . . . . . . x4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1. Some background on rigid geometry 11.1. Non-Archimedean analysis . . . . . . . . . . . . . . . . . . . . . . 11.2. Affinoid varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. Admissible coverings and rigid analytic varieties . . . . . . . . . . 31.4. The reduction of a rigid analytic variety . . . . . . . . . . . . . . 41.5. Adic topology and complete rings . . . . . . . . . . . . . . . . . . 51.6. Formal schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7. Analytification of an algebraic variety . . . . . . . . . . . . . . . 111.8. Proper morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9. Etale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.10. Meromorphic functions . . . . . . . . . . . . . . . . . . . . . . . . 141.11. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2. The structure of a formal analytic curve 172.1. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2. The formal fiber of a point . . . . . . . . . . . . . . . . . . . . . 172.3. The formal fiber of regular points and double points . . . . . . . 222.4. The formal fiber of a general singular point . . . . . . . . . . . . 232.5. Formal blow-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6. The stable reduction theorem . . . . . . . . . . . . . . . . . . . . 292.7. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3. Group objects and Jacobians 333.1. Some definitions from category theory . . . . . . . . . . . . . . . 333.2. Group objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3. Central extensions of group objects . . . . . . . . . . . . . . . . . 373.4. Algebraic and formal analytic groups . . . . . . . . . . . . . . . . 413.5. Extensions by tori . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4. The Jacobian of a formal analytic curve 514.1. The cohomology of graphs . . . . . . . . . . . . . . . . . . . . . . 514.2. The cohomology of curves with semi-stable reduction . . . . . . . 54

v

4.3. The Jacobian of a semi-stable curve . . . . . . . . . . . . . . . . 564.4. The Jacobian of a curve with semi-stable reduction . . . . . . . . 604.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A. Bibliography 71

vi

Glossary of Notations

We denote byR+ the positive real numbers as a multiplicative groupR+

0 the non-negative real numbersQ(R) the quotient field of a ring R

R the subring x ∈ R ; |x| ≤ 1 of a normed ring R

R the ideal x ∈ R ; |x| < 1 of R where R is a normed ringK a valued field

R the ring K

k the field K/KGm,K the multiplicative group of K, seen as an analytic varietyGm,K the group x ∈ K ; |x| = 1 seen as a formal analytic varietyGm,k the multiplicative group of k, seen as an algebraic varietyBnK the affinoid analytic variety SpK〈ζ1, . . . , ζn〉X, f , x the reductions of the corresponding formal analytic counterpart.

vii

Introduction

In this work, we describe the general structure of a rigid analytic curve and itsJacobian.

1. The Jacobian of a curve in the complex case

The study of the Jacobian of an algebraic curve started with the research ofcertain integrals that appear in the calculation of the circumference of an ellipse.Niels Henrik Abel and Carl Gustav Jacob Jacobi first described the Jacobianvariety around 1826. Of course, they could not formulate it in terms of algebraiccurves, since it was Bernhard Riemann, a good 25 years later, who first definedthe Riemann surface and thereby described algebraic curves over C. It tookmany more men and women to arrive at the modern description of this theoryin the middle of the twentieth century.

The Jacobian variety of an algebraic curve of genus g is the set of line bundlesof degree zero over this curve. The tensor product gives this set the naturalstructure of a group. A lot harder to show, but in the same way natural is itsstructure as an algebraic variety of dimension g itself, containing the originalcurve as a closed subvariety. The Jacobian variety is therefore an Abelianvariety with a canonical polarization, deriving from the embedding of the curve.

Over C the Jacobian variety of a curve of genus g can be described asH0(X,Ω1

X/C)′/H1(X,Z), a quotient of Cg by a lattice M of rank 2g, whichis generated by certain integrals on the curve. We can write M = Zg ⊕ ZZg,so applying the exponential function let us write Gg

m,C/ exp(2πiZ). So the Ja-cobian of a Riemann surface is the multiplicative group of C to the power gmodulo a lattice of rank g.

2. Mumford curves and general rigid analytic curves

The complex numbers are just one possibility to create a topological and alge-braic closure of Q. For every prime p we can define |x| = p−ν(x), where ν is thevaluation associated to p. This was first described by Kurt Hensel and laterrefined by his student Helmut Hasse at the end of the eighteenth century. Thetopological closure of Q with this absolute value is the field Qp of p-adic num-bers. Its algebraic closure Cp has infinite degree over Qp. There is no equivalentof the exponential function on Cp and the topology has very strange properties.Furthermore, the p-adic value gives rise to the reduction functor giving a closerelation to the finite field Fp and its algebraic closure.

The description of the Jacobian variety of a rigid analytic curve, i.e. a curveover Qp mainly decomposes into two parts, a combinatorial one and a so-called

ix

formal one. Omitting the formal part, one can describe the Jacobian of Mum-ford curves, where the reduction has a certain simple form, as Gg

m,K/M , whereM is a lattice of rank g, and thus showing a wonderful analogy to the com-plex case. Since the integral can not be defined over the p-adic numbers in ameaningful way, one needs to replace the classic formulation of the Riemannrelations with a more general, cohomological one.

To describe the Jacobian of a general rigid analytic curve, one needs to workwith Raynaud extensions, heavily researched by Michel Raynaud, SiegfriedBosch and Werner Lutkebohmert. Then one realizes that the Jacobian of arigid analytic curve can be written as E/M , where M is a lattice of rank g andE is an extension

0→ Gtm,K → E → B → 0

of a formal analytic abelian variety B by the torus Gtm,K .

3. Outline of the chapters and the results of this work

In the first chapter, we will recapitulate the basic facts about rigid analyticvarieties, their topology and their relationship with formal analytic schemesof locally topologically finite type. This chapter is by no means a completeintroduction into the topic, we refer to [BGR84] and [Bos05] for this.

In the second chapter, we will provide some new insight into the stable reduc-tion theorem, by refining the proofs of a few theorems of [BL85], using muchshorter and less technical arguments than the original work. First, we willshow that the ring OX(X+(x)) of bounded functions on the formal fiber of apoint x of the reduction is local and henselian, by taking a close look at thenormalization of x.

Secondly we will then be able to give a much more natural proof for theTheorem 2.4.1 which describes the periphery of a formal fiber. That periph-ery always consists of a disjoint union of annuli, which we can equate to thestructure of the normalization of the curve at the point x.

Finally, we will describe how this theorem is used to get to the stable reduc-tion theorem.

In the third chapter, we will introduce group objects over an arbitrary cate-gory. This allows us to form a general theory describing algebraic, rigid analyticand formal analytic groups simultaneously. With this theory, we can generalizethe results of [Ser88] and describe how an extension

0→ Gtm,K → E → B → 0

of an analytic or algebraic group B by the torus Gtm,K equals to a line bundle

over B.

The work of the third chapter pays off in the fourth and final chapter, wherewe can give the explicit description of the lattice M which E gets divided by toform the Jacobian variety of our curve. While it was known that such a latticeexists and has full rank, we can even give a constructive formula for it. It willbe shown that the formal analytic variety B does not influence the absolute

x

value of the lattice and that one gains the explicit formula for the generatorsvi = (vij)

tj=1 of the lattice as

vij =∑

en∈γi∩γj

−dn · log|qn|

where qn is the height of the formal fiber of the double point corresponding to enand dn = 1 if en has the same orientation in γi as in γj and dn = −1 otherwiseand where γi and γj are simple cycles on the curve, in close relationship to thecomplex case.

The lattice can be described fully by this method, but the description dependson the structure of the variety B of which little is known. We can construct Bexplicitly for the special family of curves X, which have a reduction consistingof a rational curve and curve Y of genus g, together with a surjective mapϕ : X → Y , with Y being a lift of Y . Then it turns out that B is isogenic toJacY of the same degree as the map ϕ.

4. Acknowledgements

This work would not have been possible without the help and support of manypeople. I would like to thank my advisor Werner Lutkebohmert for his generaland huge insight in this topic and his willingness to share it with me.

I also want to thank Stefan Wewers for being the second expert for this thesis.Many people have had part in teaching me mathematics. First and foremost

I want to thank my parents, who put their own studies at good use. Secondly,I owe my thanks to my high school teacher Herbert Langer, who managedto sneak some proofs and exactness into an otherwise quite boring and vaguecurriculum and who organized extra maths classes for the interested. Therewere many interesting courses at university, but I want to highlight the coursein analytic number theory by Helmut Maier and the courses in complex analysisand advanced algebra by Werner Lutkebohmert. These two teachers greatlyinfluenced my understanding of mathematics and finally encouraged me to writemy dissertation in this field.

I also owe thanks to Irene Bouw and to all my colleagues of the departmentfor various help in teaching, research and life in general.

Further thanks go to the chairman of the examination board Werner Kratzand his fellow examiners.

When writing a thesis in a foreign language, proofreading is even more im-portant. So therefore I want to thank my former neighbor Angela Reichmeyerfor doing this – as she put it – almost meditative work.

Writing a thesis is a quite stressful task and good friends are hard to comeby. I am very glad that I can call Katja Setzer my friend, who always had anopen ear for various discussions.

Last but by no means least I like to express my sincere thanks to my parentsCornelia and Johannes Schmieg. They brought me up, always supporting myinsatiable curiosity for the inner workings of everything.

xi

1. Some background on rigid geometry

1.1. Non-Archimedean analysis

We need the notion of a non-Archimedean normed ring.

Definition 1.1.1. A ring A together with a function |·| : A → R+0 is called a

normed ring if

1. |x| = 0 if and only if x = 0,

2. |1| = 1,

3. |xy| ≤ |x| · |y| for all x, y ∈ A,

4. |x− y| ≤ max(|x|, |y|) for all x, y ∈ A.

A normed ring is called valued if |xy| = |x| · |y| for all x, y ∈ A.

An A-module M together with a map ‖·‖ : M → R+0 over a valued ring is

called a normed A-module if the following holds

1. ‖x‖ = 0 if and only if x = 0,

2. ‖rx‖ = |r| · ‖x‖ for all r ∈ A, x ∈M ,

3. ‖x+ y‖ ≤ max(‖x‖, ‖y‖) for all x, y ∈M .

Note that valued rings are always integral and normed modules are alwaystorsion free. For any normed object T we denote by T the set x ∈ T ; |x| ≤ 1and by T the set x ∈ T ; |x| < 1. Note that T is again a normed ring/moduleand that T of a valued ring is a prime ideal in T . This leads to the definitionof T = T /T .

Definition 1.1.2. A direct sum of normed modules M = M1 ⊕M2 is calledorthonormal, denoted by M = M1 ⊥M2 if

‖(m1,m2)‖ = max(‖m1‖, ‖m2‖)

If R is a valued ring, the quotient field Q(R) is also a valued ring by defining∣∣∣ab

∣∣∣ =|a||b|

.

Proposition 1.1.3. Let A be a valued K-algebra over a valued field K with|A| = |K|. Suppose M = M1 ⊕M2 is a normed module over A, with M1 andM2 being normed modules and we further have ‖M‖ = |A|. The direct sum isorthonormal if and only if M = M1 ⊕ M2 over the ring A.

1


Proof. It is clear that we have M = M1 + M2 in any case, since the reductionis surjective. For elements m1 ∈ M1, m2 ∈ M2 the case m1 + m2 = 0 implies‖m1 +m2‖ < 1 for any lift m1,m2, but M = M1 ⊥M2. So this means we havemax(‖m1‖, ‖m2‖) < 1, i.e. m1 = m2 = 0.

For the only if part of the proof look at an element m = m1 +m2 ∈M withm1 ∈M1 and m2 ∈M2. We can assume without restriction that ‖m1‖ ≥ ‖m2‖.Suppose at first that ‖m1‖ = 1 and ‖m2‖ ≤ 1. This implies m1 6= 0. Thus weget m 6= 0 which means ‖m‖ = 1.

For the general case take any a ∈ K with |a| = ‖m1‖. Then ‖m1/a‖ = 1 and‖m2/a‖ ≤ 1. So we have ‖m/a‖ = 1 which implies ‖m‖ = |a|, thus making thesum orthonormal.

1.2. Affinoid varieties

We have two basic approaches to formal analytic varieties which both have theirmerits and flaws. In the next few sections we will explain how to construct rigidanalytic varieties and associate a reduction with them.

For this let K be a valued field, i.e. a valued ring that is a field. The associatedring of elements with value less or equal then one, is denoted by R = K andthe residue field R/R is denoted by k.

Definition 1.2.1. A power series∑k∈Nn

akζk11 ·· · ··ζ

knn is called strictly convergent

if lim|m|→∞|am| = 0. The ring of the strictly convergent power series in nvariables is called the Tate algebra Tn := K〈ζ1, . . . , ζn〉. For each f ∈ Tn wedefine the Gauss norm as |f | = max|am|.

Definition 1.2.2. The quotient A := Tn/a of Tn by some ideal a with thereduction epimorphism α is called an affinoid K-algebra. An affinoid varietySpA is the pair (MaxSpecA,A). On A we have the residue norm

|α(f)|α = infa∈a|f − a|

and the supremum norm

|f |sup = supx∈MaxSpecA

|f(x)| ,

where f(x) = f mod x for an maximal ideal x ∈ MaxSpecA as in algebraicgeometry.

It should be noted that there is always a finite field extension of K such thatthere is an epimorphism α : Tn → A with |·|α = |·|sup.

Definition 1.2.3. Let A be an affinoid K-algebra and X := SpA its affinoidvariety. For f1, . . . , fr, g ∈ A without common zeroes the subset

X(|f1/g|, . . . , |fr/g| ≤ 1) := x ∈ X ; |fi(x)| ≤ |g|, i = 1, . . . , r

is called a rational subdomain of X. If g = 1, it is called a Weierstrass domain.If ε ∈ |K×| is a constant, we write X(|f | ≤ ε) or respectively X(|g| ≥ ε) for

the corresponding rational subdomain.

2

1.3. Admissible coverings and rigid analytic varieties

Definition 1.2.4. For an affinoid variety X := SpA a subset U ⊂ X is calledan affinoid subdomain of X if there is an affinoid variety Y := SpB and amorphism ϕ : Y → X with ϕ(Y ) = U and for every morphism ϕ′ : Y ′ → Xwith ϕ′(X ′) ⊂ U there is a unique factorization ϕ′ = ψ ϕ.

A rational subdomain is an affinoid subdomain and according to Gerritzen’sand Grauert’s theorem every affinoid subdomain is a finite union of rationalsubdomains.

Proposition 1.2.5. Let X = SpA be an affinoid variety. The restriction mapOX(X(|f | ≥ ε))→ OX(X(|f | = 1)) for an f ∈ OX(X) and any 0 < ε < 1 hasa dense image.

Proof. We know that OX(X(|f | ≥ ε)) = A〈εf−1〉 and OX(X(|f | = 1)) =A〈f−1〉. By construction, the ring A[f, f−1] is dense in the latter. But A[f, f−1]is a subring of A〈εf−1〉 as well, so the image of the restriction map is dense.

1.3. Admissible coverings and rigid analytic varieties

The topology induced by a Non-Archimedean absolute value has very bad prop-erties. For example one can easily prove that such a topology is always com-pletely disconnected. To counteract this behavior one needs to introduce theconcept of a Grothendieck topology.

Definition 1.3.1. Let X be a set and S ⊂ P(X) a subset of the power set ofX. Let further be CovUU∈S be a family of coverings, i.e. a family of familieswhich satisfies Ui ⊂ U and

⋃i∈I Ui = U for every element Uii∈I ∈ Cov(U).

For a pair T = (S, CovUU∈S) the elements of S are called admissible openand the elements of Cov(U) are called admissible coverings. T is called a G-topology if it satisfies the following conditions.

1. U, V ∈ S⇒ U ∩ V ∈ S.

2. U ∈ S⇒ U ∈ CovU .

3. If U ∈ S, Uii∈I ∈ CovU and Vijj∈Ji ∈ CovUi, then the coveringViji∈I,j∈Ji is also admissible.

4. If U, V ∈ S with U ⊂ V and Vii∈I ∈ Cov V , then the covering Vi ∩Ui∈I of U is admissible.

The concept of a G-topology generalizes the usual definition of topologies.Most topological concepts, like for example continuity, can directly be trans-ferred to G-topologies by replacing open sets by admissible open ones. Tofurther add good properties one can define a unique finest slightly finer G-topology of a given G-topology as in [BGR84, pg. 338 et seqq.]. In addition tothe stated axiom, the finest slightly finer topology will satisfy the following:

(G1) Any subset V of an admissible open set U for which an admissible coveringUii∈I of U with V ∩ Ui admissible open for every i ∈ I exists will beadmissible open.

3


(G2) A covering of an admissible open set consisting of admissible open setswhich has a refinement that is admissible is already admissible itself.

As a finer G-topology, the finest slightly finer G-topology will contain X and ∅as admissible open sets if the original G-topology already did.

Definition 1.3.2. An affinoid variety X := SpA carries the weak G-topologyT which is defined by declaring the affinoid subdomains admissible open anddefining admissible coverings as finite unions of affinoid subdomains.

The finest slightly finer G-topology of T is called the strong G-topology onX. Both topologies contain ∅ and X as admissible open sets.

We can now define locally G-ringed spaces over a ring R as a set X with aG-topology and a sheaf OX of algebras over R in the usual way. Note that anaffinoid variety with the strong topology is a locally G-ringed space over K.

Definition 1.3.3. A rigid analytic variety over K is a locally G-ringed space(X,OX) over K if it satisfies the following conditions.

1. The G-topology on X contains X and ∅ as admissible open sets and hasthe properties (G1) and (G2).

2. There is an admissible covering Uii∈I of X such that (Ui,OX |Ui) is anaffinoid variety for all i ∈ I.

1.4. The reduction of a rigid analytic variety

We now want to generalize the concept of the reduction of a normed ring forrigid analytic varieties.

Definition 1.4.1. Let X = SpA be an affinoid variety. The affine schemeX = Spec A, with A = A/f ∈ A ; |f | < 1 is called the reduction of X. Themap π : X → X obtained by reducing maximal ideals is called the reductionmap.

An admissible open set U in X is called formal open if it is the preimage ofan Zariski open set under π.

Proposition 1.4.2. Let X = SpA be an affinoid variety with irreducible reduc-tion X = Spec A and |A| = |K. Then the supremum norm |·| is multiplicativeon A.

Proof. Assume that there are elements f, g ∈ A such that |f · g| 6= |f | · |g|.We can set without restriction |f | = |g| = 1, since the absolute value is alwaysmultiplicative for constants. So f, g ∈ A and have non zero reductions f and g.But A is irreducible, so f · g 6= 0 which means that |f · g| = 1 in contradictionwith the hypothesis. So |·| is multiplicative.

Proposition 1.4.3. Let X = SpA be an affinoid variety with reduction X =Spec A. Let Y = Spec B be an algebraic variety such that ϕ : Y → X is asmooth morphism and has finite presentation. There is an affinoid variety Ytogether with a morphism ϕ : Y → X reducing to ϕ.

4

1.5. Adic topology and complete rings

Proof. By our assumption, we have B = A[Z1, . . . , Zn]/(g1, . . . , gr). We takeany lift of g1, . . . , gr in A〈ζ1, . . . , ζn〉 and set Y = SpA〈ζ1, . . . , ζn〉/(g1, . . . , gr).

Definition 1.4.4. An admissible open affinoid covering Uii∈I of a rigid an-alytic variety X is called a formal covering if Ui ∩ Uj is formal open in Ui forevery i, j ∈ I.

If we are given a formal covering Uii∈I of a rigid analytic variety X we candefine the reduction X of X by reducing the Ui and gluing along the reduction ofUi∩Uj . This reduction is not unique and depends on the used formal covering.In chapter 2 we will construct coverings such that the reduction has very simplesingularities.

Proposition 1.4.5. Let X be a rigid analytic curve and U a formal coveringleading to the reduction π : X → X. Let U′ = (U ′i)i∈I be an affine coveringof X. Then U′ = (π−1(U ′i)i∈I is again a formal covering and the associatedreduction is X.

Proof. First, we need to see that U′ is admissible. Take an admissible open setUi ∈ U. Then Ui ∩ U ′j is affine for every U ′j ∈ U′. Therefore there is a finiteset of elements fi ∈ OX(Ui) such that Ui ∩ U ′j = Ui(|fi| ≥ 1) so Ui ∩ U ′j is arational subdomain of Ui and as such admissible. But then U ′j is admissible by

(G1). Using this construction we also see that U ′i is affinoid with reduction U ′i .The covering U′ is also formal since U ′i ∩ U ′j is formal by the very definition of

U′. Since the reduction of Ui ∩ U′ glues together to form Ui we get that thereduction associated to U′ is again X.


We now come to the second approach to formal analytic varieties.

Definition 1.5.1. Let R be a ring and a ⊂ R an ideal. The topology for whichak is a basis of open neighborhoods of 0 is called the a-adic topology of R.

For a module M over R we define the a-adic topology of M to be the topologywith akM as basis for 0.

The ideal a is called the ideal of definition.

The adic topology on a ring is very similar to a norm as the following propo-sition illustrates.

Proposition 1.5.2. An a-adic noetherian topological ring is normed with R =a.

Proof. Assume that R is a-adic. For x ∈ R we define

|x| =

0 if x = 0

exp(−n) where n is the smallest integer with x 6∈ an+1

5


According to Krull’s intersection theorem [Bos05, 2.1.2], we know that⋂an =

0. So this norm is well-defined and |x| = 0 implies that x = 0.If x ∈ an and y ∈ am we clearly have xy ∈ an+m and x + y ∈ amin(n,m) so|xy| ≤ |x| · |y| and |x+ y| ≤ max(|x|, |y|). Furthermore we get a = R and for nand ε with exp(−n) < ε ≤ exp(−n + 1) we have |r| < ε if and only if r ∈ an.So both induce the same topology.

The converse of this proposition is not true, as one can easily see at theexample of a normed ring where R is generated by elements of different norm.We can, however, show that valued valuation rings are always adic, as we willdo in the next few propositions.

Proposition 1.5.3. Let R be a ring with a-adic topology. Then

∞∑k=1

xk

is a Cauchy sequence if and only if (xk) is a sequence tending to zero.

Proof. The series∑∞

k=1 xk is Cauchy if for every n ∈ N we have N ∈ N such

that for every i, j > N the sum∑j

k=i xk is in an. The case i = j implies that(xk) is a zero sequence as always.

If (xk) tends to zero we find an N for every n such that xk ∈ an for k > N .But then

∑jk=i xk is in an for every i, j > N as a finite sum of elements in

an.

Definition 1.5.4. An integral ring R in which for every element x ∈ Q(R)×

either x ∈ R or x−1 ∈ R holds is called a valuation ring.

Proposition 1.5.5. Let R be an integral ring. Then the following two state-ments are equivalent:

(i) R is a valued ring with R = Q(R) = a/b ∈ Q(R) ; |a| ≤ |b|.(ii) R is a valuation ring with Krull dimension 1.

Proof. Suppose R is a valued ring with R = Q(R). We can extend the absolutevalue of R to an absolute value of Q(R) by defining∣∣∣a

b

∣∣∣ :=|a||b|

.

Then if x ∈ Q(R) we either have |x| ≤ 1 which implies x ∈ R by our assumptionor |x−1| = |x|−1 < 1 which induces x−1 ∈ R.

If p is a prime ideal not equal to zero we have x ∈ p with x 6= 0. For anyy ∈ R we can find an n ∈ N such that |y|n < |x|. Then yn/x ∈ R and we getyn ∈ p which implies y ∈ p since p is prime. Since R× = x ∈ R ; |x| = 1by the assumption that R = Q(R) this implies that p = R and R has Krulldimension 1.

A valuation ring is always local according to [Bos05, 2.1.6]. Let t ∈ R \ 0be an arbitrary non-unit. The set b = x ∈ R ; t−nx ∈ R for all n ∈ Z is

6


a prime ideal in R, since xy ∈ b, x 6∈ b implies that there is an m ∈ Z witht−mx 6∈ R and therefore tmx−1 ∈ R which implies t−ny = t−n−mtmx−1xy ∈ Rfor all n ∈ Z. Since t 6∈ b we get b = 0 by our assumption of Krull dimension1. This means that

km,x = max(n ∈ Z ; t−nxm ∈ R)

is well-defined for all x ∈ R \ 0 and m ≥ 1. Furthermore we get km,x ≥ 0.One can see directly that the limit of km,x/m exists and for every n ∈ N theinequality

kn,xn≤ lim

m→∞

km,xm≤ kn,x + 1

n

holds. We set

|x| = limm→∞

exp

(−km,x

m

).

For x ∈ R× we can calculate km,x = 0 for all m ∈ N so |x| = 1. If x is not a unit,then by the same argument as before there is an m ∈ N such that t−1xm ∈ Rand therefore |x| 6= 1. We can define km,x for every x ∈ Q(R) and one easilyrealizes that |xy−1| = |x| · |y|−1. So |·| : Q(R)× → R+ is a group morphism withkernel R× and x ∈ R if and only if |x| ≤ 1.

Any valuation ring defines a valuation with the axioms of Definition 1.1.1 ifone replaces R+ with Q(R)×/R× imbued with the canonical ordering α ≤ β ⇔αβ−1 ∈ R for α, β ∈ Q(R)×/R×. So |·| : Q(R)× → R+ makes R a valued ringwith R = Q(R).

Proposition 1.5.6. Let R be a valued ring with R = Q(R). Then the topologyof R is a-adic with a = (t) for any t ∈ R with |t| < 1.

Proof. The sets Bε := x ∈ R ; |x| < ε form a basis of the open neighborhoodsof 0 by the definition of the topology of a metric space. We need to show thatBε is open in the a-adic topology. Since |t| < 1 we find n ∈ N with |t|n < ε.Then for every x ∈ Bε and every r ∈ R we get |x + rtn| < ε and x + an is ana-adic neighborhood of x contained in Bε.

On the other hand, let x = rtn ∈ an be any element. Choose an ε < |t|n.Then |y − x| < ε implies |y/tn − r| < 1 and therefore |y/tn| ≤ 1. This meansthat there is an r′ ∈ R such that y = r′tn ∈ an and an is open.

Remark. It should be noted that even so a valued ring R with R = Q(R) is local,the ideal generating its adic topology is not the maximal ideal of R. Indeed if|Q(R)×| is dense in R+ we have Rn = R for any n ∈ N.

We can define a topology on every valuation ring that is not a field by usingall non-zero ideals as base for the topology. This topology is Hausdorffsch since0 and t are separated by any ideal generated by an element smaller than t inQ(R)×/R×. Such an element exists as t2 is smaller than t if t is not a unit andany non-unit is smaller than any unit t. If there is an ideal a in R so that thistopology of R is a-adic we call R adic.

7


Proposition 1.5.7. Let R be an adic valuation ring with a finitely generatedideal of definition. Then the ideal of definition is principal and there is a non-trivial minimal prime ideal p and the t-adic topology on R coincides with thea-adic one for every t ∈ p.

Proof. We have already given the proof for the case most interesting to us,namely when R is a valued ring. For arbitrary valuation ring the proof can befound in [Bos05, 2.1.7].

Proposition 1.5.8. Let R be a Hausdorffsch adic ring with ideal of definitiona. The topological ring R is complete if and only if

R = lim←−R/an

holds where R/an has the discrete topology.

Proof. Recall that we have

lim←−R/an = (xn) ∈

∞∏n=1

R/an ; xn ≡ xm mod am for m ≤ n .

By the definition of the inverse limit there is always a map ϕ : R → lim←−R/an.

For noetherian rings this map is injective because of Krull’s intersection the-orem, otherwise this is just the assumption that R is Hausdorffsch, so ϕ isinjective anyway.

If R is complete and x ∈ lim←−R/an is represented by a sequence (xn) ∈∏∞

n=1R/an then any sequence (xn) ⊂ R with xn = xn mod an is a Cauchy

sequence in R and its limit is x.If (xk) is a Cauchy sequence in lim←−R/a

n then for every n there is an N suchthat for every k, l ≥ N we have xk − xl = 0 mod an. We set xn = xN + an ∈R/an. Then (xn) represents an element x in lim←−R/a

n and (xk)→ x, so lim←−R/an

is complete.

Proposition 1.5.9. Let R be an adic valuation ring with ideal of definition a.The ring R = lim←−R/a

n is adic if a is finitely generated.

Proof. As said before there is an injective map ϕ : R → R. We want to showthat R is adic with ideal of definition aR. The topology on R is the coarsesttopology on lim←−R/a

n such that every projection pn : R → R/an is continuous.This means that (ker pn)n∈N is a base of neighborhoods for 0.

Let x ∈ ker pm be represented by (xn) ∈∏R/an. Then xn = 0 mod am for

every n ≥ m. If we take any lift xn ∈ R of xn we have xn ∈ am and ϕ(xn)converges to x. We have seen in Proposition 1.5.7 that we can assume a to beprincipal, i.e. a = (t). This means we can write xn = rnt

m for some rn ∈ R. Butxk ≡ xn mod ak for every n > k so rn ≡ rk mod ak−m for every n > k ≥ m.This means (rk)

∞k=m represents some r ∈ R and x = rϕ(tm). So ϕ(an)R forms

a basis of the neighborhoods of 0 and R is adic with ideal of definition aR.

Definition 1.5.10. We call R the completion of R.

8

1.6. Formal schemes

1.6. Formal schemes

With these definitions in mind, we can now explain formal schemes.

Definition 1.6.1. Let A be a complete Hausdorffsch adic ring with principalideal of definition a. For a free variable ζ we call A〈ζ〉 = lim←−A/a

n[ζ] the ringof convergent power series in ζ over A.

Proposition 1.6.2. Let A be a complete adic ring with principal ideal of defi-nition a. The elements of the ring A〈ζ〉 can uniquely be written as

∞∑k=0

akζk

with a zero sequence (ak) ⊂ A.

Proof. The ring A[ζ] is adic with ideal of definition aA[ζ] by the definition ofthe product topology. Therefore A〈ζ〉 is the completion of A[ζ]. If ak is a zerosequence then so is akζ

k and therefore∑∞

k=0 akζk is in A〈ζ〉.

If f ∈ A〈ζ〉 is any element, then there is a Cauchy sequence in A[ζ] convergingto f . This sequence gives a Cauchy sequence in A for every coefficient. But Ais complete so we can assume without restriction that the sequence for the k-thcoefficient is constant for n > Nk. Therefore we can write f =

∑∞k=0 akζ

k.

This proposition shows that the definition of K〈ζ〉 given in Section 1.2 isequal to this one.

Definition 1.6.3. Let A be a complete and Hausdorffsch adic ring with princi-pal ideal of definition a. For f ∈ A we call A〈f−1〉 = lim←−A/a[f−1] the completelocalization of A by f .

Proposition 1.6.4. Let A be a complete and Hausdorffsch adic ring with prin-cipal ideal of definition a. Then A〈f−1〉 is the adic completion of A[f−1] withrespect to the ideal aA[f−1]. There is a canonical isomorphism

R〈ζ〉/(1− fζ) →R〈f−1〉

Proof. See [Bos05, Remark 2.1.8 and 2.1.9].

Definition 1.6.5. Let A be a complete and Hausdorffsch adic ring with prin-cipal ideal of definition a. The affine formal scheme Spf A is the space of allopen prime ideals in A with Spf A〈f−1〉 as base of open sets together withthe structure sheaf A where A(Spf A〈f−1〉) = A〈f−1〉 is obtained by completelocalization.

Remark. A prime ideal p in A is open if and only if an ⊂ p for some n ∈ N. Thisimplies that p is open exactly if a ⊂ p and we get a one-to-one correspondenceof open prime ideals in A and prime ideals in A/a.

One can show (see [Bos05, 2.2]) that Spf(A) is indeed a locally topologicallyringed space, i.e. that A is a structure sheaf.

9


Definition 1.6.6. A formal scheme is a locally topologically ringed space(X,OX) such that each point x ∈ X has an open neighborhood U such that(U,OX(U)) is an affine formal scheme.

As per usual, one can construct a formal scheme by gluing affine formalschemes. With the completed tensor product, one can define a fiber product inthe category of formal schemes. See, as usual [Bos05, Chapter 2] and [BGR84,Part C] for further details.

An important case of formal schemes arise as the completion of algebraicschemes. If X is any separated scheme and Y is a closed subscheme with idealof definition J in OX , then the completion of OX alongside J together withthe topological space Y gives a formal scheme.

Definition 1.6.7. Let R be an adic valuation ring with principal ideal of defi-nition a. A topological R-algebra A is called of topologically finite type if A isisomorphic to R〈ζ1, . . . , ζn〉/b endowed with the topology induced by a.A is called admissible, if furthermore b is finitely generated and anf = 0 for

any n ∈ N implies f = 0.A formal scheme X is called admissible if there is an affine covering (Ui) of

X such that Ui = Spf Ai and Ai is admissible.

If X is an integral, projective, flat R scheme, where R is a valuation ring,then the completion of X along its special fiber gives an admissible formal Rscheme X. The scheme X is then called the analytification of X.

We want to study how admissible formal schemes are connected with rigidanalytic varieties with a fixed formal covering. For this, we introduce twocovariant functors.

Definition 1.6.8. There is a functor

rig : (admissible formal schemes over R)→ (rigid analytic varieties over K)

defined by rig(Spf A) = SpA⊗R K on the affine formal schemes, associating aformal scheme with its generic fiber, where K = Q(R).

Furthermore we have the reduction functor

red: (admissible formal schemes over R)→ (algebraic varieties over k)

defined via red(Spf A) = SpecA/a where a is the ideal of definition of R and kis the residue field of R.

To see that these functors are well-defined we again refer to [Bos05, 2.4.]. Itis easy to see that red(X) is indeed a reduction of rig(X) as this is clearly thecase if X is affine and is in general is obtained by using an admissible coveringof X. If X is an analytification of some integral, projective, flat R scheme X,then rig(X) is the analytification of the generic fiber of X as we will define itin Section 1.7.

On the other hand, if we have an affinoid variety XK = SpK〈ζ1, . . . , ζn〉/bK ,where b ⊂ R〈ζ1, . . . , ζn〉 is a finitely generated ideal and bK = b ⊗R K, thenX = Spf R〈ζ1, . . . , ζn〉/b is a formal scheme which is automatically admissible

10

1.7. Analytification of an algebraic variety

by Noether normalization. The functor rig(X) obviously yields XK and red(X)gives the canonical reduction X of X. If X is a rigid analytic variety with formalcovering (Ui)i∈I , then the covering gives gluing relations for a formal scheme,using the fact that Ui ∩ Uj is an open subset of Ui.

These relations allow us to use admissible formal schemes and rigid analyticvarieties with fixed reduction interchangeably.

1.7. Analytification of an algebraic variety

One of the most important aspects of rigid analytic varieties is the possibility toview any algebraic variety X over a non-Archimedean field K as a rigid analyticvariety over K.

We only sketch the process for affine and projective varieties. See [BGR84,9.3.4] for a more detailed discussion.

In order to construct a rigid analytic variety Xan out of the affine algebraicvariety X = SpecK[ξ1, . . . , ξn]/a we take any c ∈ K with |c| > 1 and set

Tn,k = K〈c−kξ1, . . . , c−kξn〉 .

The ring K[ξ1, . . . , ξn] is part of all the Tn,k, so we get a sequence of K-algebramorphisms

Tn,0/aTn,0 ← Tn,1/aTn,1 ← . . . ,

which gives rise to a sequence of open immersions

SpTn,0/aTn,0 → SpTn,1/aTn,1 → . . . .

By pasting along these maps we can construct an analytic variety Xan withSpTn,k/aTn,k as admissible affinoid covering. However this covering is neverformal, since SpTn,k/aTn,k yields merely a finite set of points in the reductionof SpTn,k+1/aTn,k+1.

In this work the most prominent example of an analytic variety constructedthis way is the variety Gm,K .

Proposition 1.7.1. The variety Gm,K which is the analytification of the affinevariety SpecK[ζ, ζ−1] is a rigid analytic group in the sense of Chapter 3. Theaffinoid variety Gm,K = SpK〈ζ, ζ−1〉 is an open subgroup variety of Gm,K ,consisting of the elements with absolute value 1 in K. Its reduction is Gm,k =Spec k[Z,Z−1] with a group structure induced by Gm,K .

Proof. Since SpecK[ζ, ζ−1] is the algebraic description of Gm,K and morphismscarry over in the analytification process we see that Gm,K is the analytic groupvariety with the group structure of K×. The variety Gm,K is the first varietyof the defining sequence of Gm,K and as such Gm,K is an open subgroup ofGm,K .

If X is projective, say X ⊂ Pn, then there are (n+ 1) affine

Ui = SpecK[ζ0/ζi, . . . , 1, . . . , ζn/ζi]

11


in Pn for which we can find analytic counterparts. The gluing of the Ui is doneby equating ζk/ζi with ζk/ζj ·ζj/ζi where ζi/ζj and ζj/ζi are not null. But thesegluing relations are already defined for the affinoid varieties

SpK〈ζ0/ζi, . . . , 1, . . . , ζn/ζi〉

where ever |ζi/ζj | = 1, so the analytification of Pn and the subset X can beobtained by gluing n+ 1 copies of BnK at their borders.

Since BnK is affinoid with reduction Ank and we glued at

SpK〈ζ0/ζi, . . . , 1, . . . , ζ0/ζi, ζi/ζj〉

which reduces to the correct gluing for Pnk , this also gives a formal covering ofX.

1.8. Proper morphisms

We want to give the definition of a proper analytic variety and show that theanalytification of a projective algebraic variety is proper. To be able to do so, weneed to formulate the concepts of separate morphisms and relatively compactsubsets.

Definition 1.8.1. A morphism ϕ : X → Y of analytic varieties is called sepa-rated, if the diagonal morphism ∆: X → X ×Y X is a closed immersion.

One easily realizes as in [BGR84, 9.6] that morphisms of affinoid varietiesand therefore affinoid varieties over SpK are always separated. Since the defi-nition coincides with the algebraic one, it is easy to see that analytifications ofseparated algebraic morphisms are again separated.

Definition 1.8.2. Let X = SpA and Y = SpB be affinoid varieties with amorphism ϕ : X → Y . An affinoid subset U ⊂ X is said to be relatively compactin X over Y if there exists an affinoid generating system f1, . . . , fr of A over Bsuch that U ⊂ X(|f1| < 1, . . . , |fr| < 1).

The relative compactness of a subset U ⊂ X is equivalent of assuming thatthere is an ε ∈

√|K×| with ε < 1 such that U ⊂ X(ε−1f1, . . . , ε

−1fr).Now we can define proper morphisms.

Definition 1.8.3. A morphism ϕ : X → Y of analytic varieties is called properif ϕ is separated and if there is an admissible affinoid covering (Yi)i∈I of Y suchthat, for every i ∈ I there are two finite admissible affinoid coverings Xij andX ′ij of ϕ−1(Yi) such that Xij is relatively compact in X ′ij over Yi for all indicesi and j.

Proposition 1.8.4. The analytification of a projective variety is proper.

Proof. The closed ball BnK is relatively compact in every ball of greater ra-dius. Since a projective variety allows finite admissible coverings coming fromBnK with any radius greater or equal to one we get the necessary covering forproperness.

12

1.9. Etale morphisms

1.9. Etale morphisms

Definition 1.9.1. Let ϕ : X → Y be a morphism, x ∈ X a point with anadmissible open neighborhood U and an immersion ι : U → BnY . The morphismϕ is called smooth of relative dimension r at x if there are n − r sectionsg1, . . . , gn−r locally at y = ϕ(x) generating the ideal defining U as a subschemeof BnY with dg1, . . . ,dgn−r being linearly independent in Ω1

BnY /Y.

Furthermore ϕ is called formal smooth of relative dimension r if the differen-tial forms dg1, . . . ,dgn−r form an orthonormal system in Ω1

BnY /Y.

Definition 1.9.2. A morphism is called etale or formal etale if it is smooth orformal smooth of relative dimension 0 respectively.

Definition 1.9.3. Let R be a local ring with residue field k. The ring R iscalled henselian if any monic polynomial p ∈ R[T ] which admits a factorizationp = f · g ∈ k[T ] with coprime factors f and g in the reduction has lifts of f andg in R[T ] with p = f · g.

The smallest local ring extension Rh of R such that Rh is henselian is calledthe henselization of R.

We mainly need the characterization of the henselization by etale morphismsThis is done by the next proposition.

Proposition 1.9.4. Let R be a local ring. The henselization Rh of R is thedirect limit lim−→i∈I Ri of all isomorphism classes of R-algebras Ri which occuras local rings of some etale R-scheme at the closed point lying over the closedpoint of R.

See [BLR90, 2.3]The last proposition allows us to assume that the local ring OX,x is henselian

if one allows for etale base change.

Proposition 1.9.5. Let X be an algebraic curve over an algebraically closedfield k and let x ∈ X a closed point. There is an etale morphism X → X suchthat x lies on n different irreducible components of X, where n is the numberof points in the fiber of x in the normalization.

Proof. If x lies on n different irreducible components, then the normalizationof X has n disjoint components with points lying over x. Since according to[Ray70, p. 99] henselization commutes with normalization we can assume thatx lies on exactly one irreducible component and show that there is only onepoint in the normalization of X over x.

Since we look at X up to etale morphism we can assume the local ring OX,xto be henselian. Let y1, . . . , yn be the points lying over x in its normalizationX ′. There are gi ∈ OX′,y1,...,yn such that p = (T −g1) · · · · · (T −gn) ∈ OX,x[T ] isa monic polynomial and that gi(yj) = 1−δij . Then p(x) = T ·h where h ∈ k[T ]is not divisible by T since we can calculate p(x) as p(yi) for any i between 1and n. But OX,x is henselian and any non trivial factorization of p is contraryto our assumptions, which implies n = 1.

13


The lift of a morphism ϕ as constructed in Proposition 1.4.3 is etale if ϕ isetale.

Proposition 1.9.6. Let X and Y be affinoid varieties with associated reduc-tions πX : X → X and πY : Y → Y . Let Y be formal smooth over K. For everymorphism ϕ : X → Y there is a lift ϕ : X → Y of ϕ.

Proof. We set X = Spf A and Y = Spf B, A = lim←−A ⊗ Ri, B = lim←−B ⊗Ri, R = lim←−Ri with Ri = R/ai as before. By [BLR90, Prop. 2.2.6] we getHomR(Xi, Y )HomR(Xi−1, Y ) since Y → Spf R is smooth. Therefore themap ϕ, which gives a map ϕ1 : X1 → Y successively lifts to maps ϕi : Xi → Yand therefore ϕ : X → Y exists and has the proposed reduction.

Proposition 1.9.7. Let X and Y be affinoid varieties with reductions X andY . Let ϕ : X → Y be a formal smooth morphism admitting a section σ : Y → Xin the reduction. There is a lift of σ which is a section of ϕ.

Proof. Again by [BLR90, Prop. 2.2.6] we get HomY (Yi, X)HomY (Yi−1, X),which lifts σ to σ ∈ HomY (Y,X). By definition of HomY , σ is a section ofϕ.

1.10. Meromorphic functions

Definition 1.10.1. Let X be a reduced rigid analytic variety and U an openaffinoid subvariety. The field of fractions Q(OX(U)) is called the field of mero-morphic function over U . It extends to a sheaf MX of meromorphic functionson X. The global sections of this sheaf are called meromorphic functions.

Proposition 1.10.2. Let X be a projective rigid analytic curve over an alge-braically closed field K and U be an affinoid subvariety with irreducible reduc-tion. Then the valuation of OX(U) extends canonically to MX(U).

Proof. Let f ∈ MX(U) be a meromorphic function. There are functionsg, h ∈ OX(U) with f = g/h by definition. We set |f | = |g|/|h|. Accordingto Proposition 1.4.2, the absolute value on U is multiplicative and as such welldefined.

We can define a reduction of a meromorphic function f if |f | = 1 on Xi =π−1(Xi \ Sing X) by setting f = g/h which is a rational function on Xi.

Proposition 1.10.3. Let X be a projective smooth rigid analytic curve withreduction X. Let f ∈ k(X) be a rational function. Then f has a meromorphiclift f ∈MX(X).

Proof. Let f be defined on U ⊂ X. Let U := π−1(U), then we find a liftf ∈ OX(U) of f . The algebraic functions are dense in OX(U) by definitionwhich lets us approximate f with algebraic functions on U and therefore alge-braic rational functions on X. The limit of these functions therefore gives ameromorphic lift of f .

14

1.11. Examples

Proposition 1.10.4. Let X be a projective rigid analytic curve and let f ∈MX(X) be a non-constant meromorphic function. The set x ∈ X ; |f(x)| ≤1 is an affinoid variety.

Proof. The meromorphic function f : X → P1 defines a finite covering map ofX. Since B1 = x ∈ P1 ; |x| ≤ 1 and x ∈ P1 ; |x| ≥ 1 is an admissibleaffinoid covering of P1, the preimages of these sets are affinoid too.

1.11. Examples

Example 1.11.1. The unit ball BnK = SpK〈ζ1, . . . , ζn〉 = x ∈ Kn ; |xi| ≤ 1is an example for an affinoid variety. The unit ball reduces to the affine spaceAnk .

Example 1.11.2. The affine space AnK can be seen as a rigid analytic varietyby covering it with balls with increasing radii. Note that any ball of strictlysmaller radius in a bigger ball reduces to a point, so this covering is not formal.

Example 1.11.3. The projective space PnK can be obtained by the usual wayof covering it with n + 1 copies of AnK . One sees that covering it with n + 1copies of the unit ball which are identified by their border x ∈ K ; |x| = 1 isalready enough. This covering is formal and the reduction obtained this way isthe Pnk .

A projective algebraic variety given by homogeneous equations in some PnKtherefore also admits a formal covering and its reduction according to thiscovering can be computed by reducing the equations after normalization. Wedenote such a rigid analytic variety as a projective analytic variety.

Example 1.11.4. Let q ∈ Gm,K with |q| < 1. The elliptic curve with parameterq is defined as the quotientX = Gm,K/(q

Z). We want to construct an admissibleaffinoid covering of X. Let

U1 := x ∈ Gm,K ; q ≤ |x| ≤ √q, U2 := x ∈ Gm,K ;√q ≤ |x| ≤ 1 ,

two annuli of height√q. We can map Ui onto X by taking the points modulo

q. Both maps are analytic isomorphisms. Thus U = U1, U2 is an admissibleaffinoid cover of X. Since

U1 = SpK〈√q−1ζ, qζ−1〉

and

U2 = SpK〈ζ,√qζ−1〉

we get

Ui = Spec k[Xi, Yi]/(Xi · Yi)

mapping√q−1ζ to X1, qζ−1 to Y1, ζ to X2 and

√qζ−1 to Y2. The relation

q ≡ 1 gives us the gluing isomorphisms X1 = 1/Y2 and X2 = 1/Y1. Thus Xhas two components of genus 0, meeting in two double points.

15


The reduction map π maps points with absolute value between q and√q

to the double point of U1, points with absolute value between√q and 1 to

the double point of U2. Points with absolute value 1 are mapped onto thecomponent X1 = Y2 = 0 and those with absolute value

√q are mapped onto

the component X2 = Y1 = 0. In Chapter 4 we will show that this curve isindeed an elliptic curve.

16

2. The structure of a formal analyticcurve

In this chapter, we will analyze the structure of a p-adic curve and its depen-dence on the structure of its reduction.

2.1. Basic definitions

Definition 2.1.1. Let K be an algebraically and topologically complete non-Archimedean valued field. An analytic variety X over K of dimension 1 with afixed reduction π : X → X is called a formal analytic curve if X is smooth overK. Therefore OX(U) is a reduced and irreducible K algebra of dimension 1.

The first important to mention fact about analytic curves is that accordingto [BGR84, 6.4.3/1], over an algebraically closed field K, we can assume thatthe residue norm on an affinoid subdomain of X coincides with the supremumnorm. The supremum norm is power-multiplicative so the reduction can notcontain any nilpotent elements. Furthermore the reduction is of finite type overk, so the reduction is a reduced curve. This means the reduction always consistsof a finite union of reduced and irreducible components.

2.2. The formal fiber of a point

Let X/K be an analytic curve with reduction π : X → X.

Definition 2.2.1. The preimage of a point x ∈ X under π is called the formalfiber X+(x) of x. An admissible formal affinoid variety U ⊂ X with π−1(x) ⊂ Uis called a formal neighborhood of the formal fiber.

We want to discuss how the kind of the singularity of a point of the reductiondetermines the formal fiber.

Proposition 2.2.2. Let X = B1 = SpK〈ζ〉 be the unit ball with the canonicalreduction X = A1

k = Spec k[Z]. Then x ∈ B1 ; |ζ(x)| < 1 is the formal fiber

of 0 ∈ X. Furthermore OX(X+(x)) is local and the reduction map extends to

OX(X+(x))→ OX,x. Furthermore for every f ∈ OX we have f(x) = f(x).

Proof. We write x for the zero point of X = A1k. Let x ∈ X+(x) be a point in

the formal fiber with associated maximal ideal mx = (f) for a function

f =∞∑k=0

akζk ∈ R〈ζ〉 .

17

2. The structure of a formal analytic curve

Since f generates the ideal (Z) corresponding to the point x we can assumewithout restriction that a0 = 0 and a1 = 1 and therefore we get |a0| < 1 and|a1| = 1. This means that

|ζ(x)| =

∣∣∣∣∣∣ 1

a1

∞∑k=0,k 6=1

akζ(x)k

∣∣∣∣∣∣ ≤ |a0| < 1 .

So x is in x ∈ B1 ; |ζ(x)| < 1.On the other hand the function f = ζ − x will reduce to Z for any x in the

given set and therefore (f) will generate a maximal ideal in the formal fiber.This means that OX(X+(x)) is obtained from

lim←−β<1

OX(X)〈β−1ζ〉 = RJζK .

A series f =∑∞

k=0 akζk is a unit in RJζK if and only if |a0| = 1, which means

that the sum of two non-units is again not a unit and that OX(X+(x)) is local.Furthermore, for any x ∈ X+(x) we have |f(x)| < 1 if and only if |a0| < 1

and therefore f(x) = 0. Since we can use a linear transformation to move any

point into the origin we get f(x) = f(x) for every x ∈ B1.

Proposition 2.2.3. Let X = SpA be an analytic curve with smooth reductionX. Then X+(x) is isomorphic to x ∈ B1 ; |ζ(x)| < 1 for any x ∈ X.

Especially OX(X+(x)) is local and f(x) = f(x) for every f ∈ OX .

Proof. We adapt the proof of [BL85, Prop. 2.2] to our purposes. We choosea point x ∈ X with π(x) = x. Since x and x are regular, we can localize Ain a way that there is a function f with absolute value 1 which generates themaximal ideal corresponding to x and its reduction will generate the maximalideal corresponding to x. Now we get

A = K ⊕Af and A = k ⊕ Af .

And therefore we getA = K ⊥ Af (2.1)

according to Proposition 1.1.3. We define the morphism σ : K〈ζ〉 → A whichmaps ζ to f .

Look at

σ : K〈ζ〉 _

// A _

, ζ 7→ f

σε : K〈ε−1ζ〉 // A〈ε−1f〉 , ζ 7→ f .

The map σε is injective for all ε since the map f : X → B has zero dimensionalfibers. Furthermore for any element g ∈ A we set

g0 = g and gi = gi(x) + gi+1 · f

18


according to (2.1). So we get |gi| ≤ 1. Then the series

∞∑i=0

gi(x)f i =∞∑i=0

gi(x)εi(ε−1f

)iconverges on A〈ε−1f〉 and is in the image of σε. But then every series in ε−1fis also in the image of σε and therefore σε is surjective.

If we have any g ∈ OX such that g ∈ mx we know that g = hf and thereforethat |g − h · f | < 1 for any lift h ∈ OX of h. This means |g(x)| < 1 for any xwith |f(x)| < 1 and so the formal fiber is composed of all x ∈ X which fulfill|f(x)| < 1.

This gives us the necessary isomorphism of the formal fiber.

With these two propositions, we can describe the formal fiber of a point witha function f ∈ OX . If X is non-singular everywhere but in the interesting pointx and f has a single isolated zero at x, we now know that |f(y)| = 1 for anypoint outside the formal fiber. So we get x ∈ X ; |f(x)| < 1 ⊂ X+(x). Wewant to prove that equality holds.

Proposition 2.2.4. Let X = SpB be an analytic curve with reduction X. Letfurther be f ∈ OX(X) be a function so that f has a single isolated zero x and letX \x be non-singular. Let further be g ∈ OX′ a function on the normalization

X ′ of X. Then there is an ε < 1 and a lift g ∈ OX(Xε) where Xε = X(|f | ≥ ε).

Proof. Let us first fix some notation. We set A := K〈ζ〉 and B1K = SpA. We

can look at f as a function f : X → B1K with a corresponding injection A →B

which maps ζ to f . Let X := Spec B, Ak := Spec A and X ′ := Spec B′, thelatter being the normalization. We will further need the varieties

Xε,s := x ∈ X ; ε ≤ |f(x)|, |s(x)| = 1

and

B1ε,s := x ∈ B1

K ; ε ≤ |ζ(x)|, |s(x)| = 1 ,

and their corresponding affinoid algebras Bε,s = B〈εf−1, s−1〉 for Xε,s andAε,s = A〈εζ−1, s−1〉 for Bε,s where s is a function in OB with s(0) 6= 0 chosenso that f is finite.

SinceB0,s is finite over A0,s the quotient fieldQ(B0,s) is algebraic overQ(A0,s)and Q(B) = Q(B′) is algebraic over Q(A). Furthermore B0,s is the integralclosure of A0,s and B′s is the integral closure of As in their respective fields.

Since K is algebraically closed, K is stable. Due to [BGR84, 5.3.2 Thm.1] Q(K〈ζ〉) = Q(A) is stable. Since B/A is generically unramified, we getaccording to [BGR84, 3.6 Prop. 8] that

t := [B0,s : A0,s] = [Bs : As] . (2.2)

The given element g induces a function on X \ x so there is a lift g ∈OX(X \X+(x)). The restriction map OX(ε ≤ |f |)→ OX(1 ≤ |f |) has a denseimage so we can choose g ∈ OX(ε ≤ |f |) = Bε without restriction.

19


We have already seen that B0,s is integral over A0,s and therefore Bεf−1,s isintegral over Aεζ−1,s. Completion gives that Bε,s is integral over Aε,s so thereis an integral equation

F (T ) := (−1)tT t + a1Tt−1 + · · ·+ at

of degree t, that annuls g with ai ∈ Aε,s which is the characteristic polynomialof g.

Let

ai =∑ν∈Z

aiνζν

be the Laurent series representation of ai. Because of (2.7) the reduction F ofF is the characteristic polynomial of g. Therefore we have ai ∈ As and thus|aiν | < 1 for all ν < 0. Furthermore since ai ∈ Aε,s the sequence aiνε

ν tends tozero for ν → −∞. So there is an N0 such that |aiνεν | < 1 for all ν < N0. Since|aiν | < 1 for all ν < 0 we can find ε1 < 1 such that |aiνεν1 | < 1 for N0 ≤ ν < 0.Thus, by adjusting ε, we have |aiνεν | < 1 for all ν < 0. Therefore |ai| ≤ 1 andthus |g| ≤ 1 on Xε,s.

Proposition 2.2.5. Let X, f and g be as in Proposition 2.2.4. Denote the zeroof f by x. There is a point x ∈ Xε = X(ε ≤ |f | < 1) such that |g(x)| = 1 ifand only if there is a point y ∈ X ′ lying over x with g(y) 6= 0.

Proof. It is important to note that by Proposition 2.2.3 we know that |f(x)| <1 already implies that x is in the formal fiber. Furthermore we know that|ak(x)| < 1 for one x with |f(x)| < 1 implies that |ak(x)| < 1 for any x with|f(x)| < 1 since ak ∈ Aε.

Since x is the only zero of f we know that g has m zeroes on the n pointsy1, . . . , yn lying over x if and only if ak(0) = 0 for k < m and am(0) 6= 0. So|ak(x)| < 1 for k < m and |am(x)| = 1 for any x in the periphery of the formalfiber. Therefore there is a point x with |g(x)| = 1. The other implication followsby the same argument.

Lemma 2.2.6. Let X be an analytic curve and x ∈ X a point of the reduction.Then the ring OX(X+(x)) is local with residue field k.

Proof. We can, without restriction, assume that X = SpB and X = Spec B.We can further assume that X has no singularities beside x.

Let h1, h2 ∈ OX(X+(x)) be two non-units. This means that there are pointsx1, x2 ∈ X+(x) such that |h1(x1)|, |h2(x2)| < 1.

If |h1(x2)| < 1 then |(h1 + h2)(x2)| < 1 and h1 + h2 is not a unit. Thereforewe can assume that |h1(x2)| = 1.

Let us first move back in an affinoid case. Set f = h1 · h2 and choose β < 1such that |f(x1)|, |f(x2)| < β. We set Bβ = B〈f/β〉 with reduction Bβ. Since

h1 ∈ OX(X+(x)) we get h1 ∈ Bβ. By the choice of β the function h1 is still nota unit in Bβ but |h1| = 1 in Bβ.

The function h1 is defined on Bβ and therefore is defined on the normalizationof Xβ as well. Furthermore, since |h1(x1)| < 1 we get h1(x) = 0, so h1(y) = 0 for

20


all y lying over x in the normalization. This means |h1(x2)| < 1 by Proposition2.2.5, contrary to our assumption.

Therefore an element of h ∈ OX(X+(x)) is not a unit if and only if |h(x)| < 1on every point x ∈ X+(x). This implies that the sum of two non-units is againnot a unit and OX(X+(x)) is local.

Corollary 2.2.7. Let X be an analytic curve and h ∈ OX(X+(x)) be a func-tion. If |h(x)| < 1 for any x ∈ X+(x) then |h(x)| < 1 for all x ∈ X+(x).

Corollary 2.2.8. Let X be an analytic curve and f ∈ OX be a function suchthat f has a single isolated zero in the point x. Then X+(x) = X(|f | < 1).

Proof. As we have seen |f(x)| < 1 on every point of X+(x).

Corollary 2.2.9. Evaluating functions and taking their reduction commutes.

We get f(x) = f(x).

Lemma 2.2.10. Let X be a analytic variety and x ∈ X a point of the reduction.Then OX(X+(x)) is henselian.

Proof. Let us set B+ := OX(X+(x)). Let P = Tn+cn−1Tn−1+· · ·+c0 ∈ B+[T ]

be a monic polynomial with reduction P ∈ k[T ] which has a simple zero α ink. After coordinate transformation we can assume that α is 0. We denote theother zeroes by αi with i = 2, . . . , n, where n is the degree of P .

In this setup we see that c0 = 0 and therefore c0 ∈ m, the maximal ideal ofB+. Since c1 is the sum of every possible product of n − 1 zeroes of P andonly one such combination, namely α2 · · · · · αn differs from zero, we know thatc1 6∈ m.

For every x ∈ X+(x) and |ε| < 1 the polynomial P can be written as

P = εn · Tn/εn + cn−1(x)εn−1 · Tn−1/εn−1 + · · ·+ c1(x)εT/ε+ c0(x)

Now c0 ∈ m and c1 6∈ m imply |c0(x)| = η < 1 and |c1(x)ε| = |ε| by Corol-lary 2.2.7. Furthermore for i > 1 we get |ci(x)εi| ≤ |εi| < |ε|, so P is T/εdistinguished of order 1 if ε > η.

According to the Weierstraß preparation theorem [Bos05, Sect. 1.2 Cor. 9]this that means there is a unit ωε in K〈T/ε〉 and an element αε(x) ∈ K suchthat

P (x) = (T/ε− αε(x)) · ωε(x) = (T − αε(x) · ε) · ωε(x)/ε . (2.3)

Since P is a polynomial and ωε is a convergent power series, ωε can only be apolynomial itself.

The uniqueness of Weierstraß preparation assures us that α(x) := αε(x) · εdoes not depend on ε and as such gives a function α ∈ B+. Furthermore,we know that |T/ε − αε(x)| = 1 in the absolute value of K〈T/ε〉, so |α(x)| =|αε(x) · ε| ≤ |ε| < 1 which means that α is in m, i.e. α is the lift of α we lookedfor.

Corollary 2.2.11. The ring of functions on a formal fiber of a point is invari-ant under etale base change.

21


2.3. The formal fiber of regular points and double points

Proposition 2.3.1. Let X = SpA be an analytic curve with reduction X. Apoint x ∈ X of the reduction is regular if and only if the formal fiber X+(x) isisomorphic to x ∈ B1 ; |x| < 1.

Proof. We have already seen in Proposition 2.2.3 that X+(x) is isomorphic tothe open unit ball if x is regular.

If we suppose that X+(x) is isomorphic to x ∈ B1 ; |x| < 1 we know by

Lemma 2.2.6 that ˆA is isomorphic to kJT K and therefore that x is regular.

Proposition 2.3.2. Let X = SpA be an analytic curve with reduction X. Apoint x ∈ X of the reduction is an ordinary double point if and only if theformal fiber X+(x) is isomorphic to x ∈ B1 ; ε < |x| < 1, an open annuli inB1.

Proof. At first we want to reduce to the case that the double point lies on twodifferent components. This can be done by applying an etale base change sothis is simply a consequence of Corollary 2.2.11. However, the same result canbe obtained more directly. This serves to illustrate our proof of said corollary.

Assume the maximal ideal mx corresponding to x is generated by two func-tions g1, g2 such that f := g1 · g2 ∈ m3

x. This is possible since x is a doublepoint. Assume further that without restriction ordyi f = 3 on both points yilying over x in the normalization. This assumption can easily be met by havingordyi gi = 1 and ordyi gj = 2 for i, j = 1, 2, i 6= j.

Now look at hi := g3i /f for some lifts gi, f of gi, f . Their reduction is defined

on the normalization of X and so hi is defined and of absolute value smallerthan 1 on X(|f | ≥ ε3) for a suitable ε. But we have hi(yi) 6= 0 so there is apoint xi ∈ X+(x) with |hi(xi)| = 1. So we can set β such that |f(xi)| = β3 < 1and we get |gi(xi)|3 = |f(xi)| which means that |gi/β| = 1 on X(ε3 ≤ |f | ≤ β3).Note that by adjusting ε we can get values for β arbitrarily close to 1. Thismeans that on A〈β−3f〉 we have g1/β · g2/β = f/β2. But |f | = β3 on A〈β−3f〉so this reduces to g′1 · g′2 = 0. Therefore x joins two components of the reductionof X(|f | ≤ β3) for any β < 1 and we only need to discuss this case.

We will use the proof as in [BL85, Prop. 2.3]. Let mx be generated by fand g such that f · g = 0. Assume that f and g are lifts chosen so that theyhave a common zero x. This can be achieved by replacing g with g − g(x) fora zero x of f . Since |g(x)| < 1 the reduction g is left unchanged. Therefore mx

is generated by f and g and we get

A = K ⊥ Af ⊥ Ag (2.4)

as in the case of a regular point. We recursively define sequences (fi), (gi), (hi)in A and (αi) in K. For this set f0 = g0 = α0 = 0 and define

hk :=

(f −

k−1∑i=0

fi

)(g −

k−1∑i=0

gi

)−k−1∑i=0

αi (2.5)

22

2.4. The formal fiber of a general singular point

Then we decompose hk according to (2.4) to get

hk = αk + gkf + fkg (2.6)

Using (2.6) in (2.5) gives us

hk = fk−1

k−2∑i=0

gi + gk−1

k−2∑i=0

fi + fk−1gk−1

for k ≥ 2. By our assumption of f and g we get |h1| = |f ·g| = γ < 1. Thereforewe recursively get |fk|, |gk|, |αk| ≤ |hk| ≤ γk. Set

f ′ :=

(f −

∞∑i=1

fi

)and g′ :=

(g −

∞∑i=1

gi

)

Then f ′ ·g′ =∑∞

i=1 αi =: α ∈ K. Since f ′ = f and g′ = g we know that |c| < 1.Furthermore c 6= 0 since A is an integral domain.

We define

σ : K〈ζ, cζ−1〉 → A

ζ 7→ f ′

cζ−1 7→ g′

and again set σε : K〈ε−1ζ, ε−1cζ−1〉 → A〈ε−1f ′, ε−1g′〉 for the induced map.We know that σ is injective because of (2.4). For any element h ∈ A wecan construct a series h = h0 +

∑∞k=0 hkf

′k +∑∞

k=0 hkg′k with hk ∈ K by

repeated application of (2.4). The series converges as long as |ε| < 1 so σε is anisomorphism in this case.


Theorem 2.4.1. Let X = SpB be of pure dimension 1, and let x be a pointin X. Let U be a formal neighborhood of X+(x) in X and let f be a functionin OX(U) such that x is an isolated zero of f ∈ OX(U). Let x′1, . . . , x

′n be the

points in the normalization X ′ of X lying over x. Then, for ε ∈ |K×|, ε < 1close to 1, the analytic variety x ∈ X+(x) ; ε ≤ |f(x)| < 1 decomposes inton connected components R1, . . . , Rn which are semi-open annuli.

More precisely, let ζ be a coordinate function of B1, and, for i = 1, . . . , n,denote by ti = ordy′i(f) the vanishing order of f at x′i. There are isomorphisms

ϕi : Ri −→z ∈ B1 ; ε1/ti ≤ |ζ(z)| < 1

such that, up to a unit in OX(Ri), the function f |Ri equals the pullback ϕ∗i (ζti).

Furthermore, if the reduction h ∈ OX,x of an element h ∈ OX(X+(x)) satis-

fies τ = ordx′i0(h) < ∞ for some index i0, and if ε is close enough to 1, then,

up to a unit in OX(Ri0), the function h|Ri0 equals the pullback ϕ∗i0(ζτ ).

23


Proof. Let us first provide a proof for the simpler situation where f only hasone isolated zero x in X and only one point x′ lies above x in the normalization.

The first step of the proof is to use Lemma 2.2.4 on a local parameter of x′.Since we have not assumed X \ x to be non singular, we need to adjust theproof a little.

Let us recall some notation. Let X = SpB and B = SpA with A := K〈ζ〉.We have, as usual, the morphism f : X → B corresponding to an injectionA →B which maps ζ to f . Let X := Spec B, A := Spec A and X ′ := Spec B′,the latter being the normalization. We will further need the varieties

Xε,s := x ∈ X ; ε ≤ |f(x)|, |s(x)| = 1

and

Bε,s := x ∈ B ; ε ≤ |ζ(x)|, |s(x)| = 1 ,

where s is a function in OB and s(0) 6= 0 yet to be determined fully but chosenso that f is finite. Note that Xε,s and Bε,s only depend on ε and the reductions of s.

Since K is algebraically closed, K is stable. Due to [BGR84, 5.3.2 Thm.1] Q(K〈ζ〉) = Q(A) is stable. Since B/A is generically unramified, we getaccording to [BGR84, 3.6 Prop. 8] that

t := [B0,s : A0,s] = [Bs : As] . (2.7)

Let g be a local parameter of x′ in X ′. Then we have Q(B′) = Q(A)[g]. Wecan choose s as desired so that localizing at ζ and s yields Bζ·s = Aζ·s[g], wheres(0) 6= 0.

We can remove singularities by localizing, so we can apply Lemma 2.2.4 toget an ε such that a lift g of g is defined on Xε,s.

The polynomial F of the lemma is the minimal polynomial of g up to nor-malization, by our choice of assumptions.

Next, we want to show that

Bε,s = Aε,s[g] .

Since Q(B)/Q(A) is a vector space, we can adjust s in a way that Bs is even afinite free As module, namely

Bζ,s = Aζ,sg0 ⊕ · · · ⊕ Aζ,sgt−1 .

By applying Proposition 1.1.3 we get

B1,s = A1,sg0 ⊥ · · · ⊥ A1,sg

t−1 (2.8)

i.e. g0, . . . , gt−1 are an orthonormal system of generators of B1,s over A1,s.

Furthermore, since F is a Weierstraß-polynomial, we can decompose A[g]according to [BGR84, 5.2.3 Prop. 3] so that

A[g] = Ag0 ⊕ · · · ⊕Agt−1

24


and alsoAε,s[g] = Aε,sg

0 ⊕ · · · ⊕Aε,sgt−1 (2.9)

Since B1,s = A1,s[g] and the support of the quotient Bε,s/Aε,s[g] consists offinitely many points. So there exists an ε < 1 such that Bε,s = Aε,s[g].A(ζ) → B′(f) is a finite extension of discrete valuation rings, with ζ being a

local parameter in the former and g in the latter. Therefore there is a unit uso that ζ = gtu. Since the residue field is algebraically closed, we can, withoutloss of generality, write u = 1 + δ with δ ∈ m. So we get the representation

ζ = gt(1 + δ) (2.10)

with |δ(x)| < 1 where for x ∈ Xε,s and |ζ(x)| < 1.Now δ has a series representation

δ = δ0g0 + · · ·+ δt−1g

t−1

with δi ∈ Aε,s,δi =

∑ν∈Z

diνζν .

Due to (2.8) we have |diν | ≤ 1. Due to (2.9) we have diνεν → 0 for ν → −∞

and so there is an N0 such that |diνεν | < 1 for all ν < N0. We claim |di,ν | ≤ 1for all ν ∈ Z and |di,νεν | < 1 for all ν ≤ 0.

We know |δi| ≤ 1 and δi ∈ Aε,s. Since δ ∈ m and

ˆB′ = ˆAg0 ⊕ · · · ⊕ ˆAgt−1

we must have δi ∈ A(ζ) ⊂ kJζK. As in the lemma above, we can modify ε ifnecessary to get that |diνεν | < 1 for all ν < 0. The case ν = 0 was not coveredabove, however, this holds since δ ∈ m. This shows that

δ ∈ R〈ζ, ε/ζ〉 · g0 ⊕ · · · ⊕R〈ζ, ε/ζ〉 · gt−1 (2.11)

holds.On the other hand, by the same reasoning, we also have a γ ∈ Bε,s with

gt = ζ(1 + γ) .

Yet again, we can write

γ = γ0g0 + · · ·+ γt−1g

t−1

withγi =

∑ν∈Z

ci,νζν

and the reduction γi ∈ kJζK. So we have |ci,ν | ≤ 1 for all ν ∈ Z and |ci,νεν | < 1for ν ≤ 0. Again, this means that

γ ∈ R〈ζ, ε/ζ〉 · g0 ⊕ · · · ⊕R〈ζ, ε/ζ〉 · gt−1 (2.12)

25


Choosing an element α ∈ K with αt = ε, we get

ε

ζ=

(α

g

)t· (1 + γ) (2.13)

We want to show that for any arbitrary number β with |β| < 1 we have

R〈g/β, α/g〉 = R〈ζ/βt, ε/ζ〉[g]

and that therefore X+(x) is isomorphic to a semi-open annulus.Let w ∈ R be a number with |w| = max(|δ|, |γ|) where the absolute value of

the ring R〈ζ/βt, ε/ζ〉[g] is used. Our calculations above have shown that

|w| ≤ max

supi=1...t−1ν≤0

(di,νεν) , sup

i=1...t−1ν≤0

(ci,νεν) , βt

< 1 .

By (2.10) and (2.13) we have

ζ, ε/ζ ∈ R〈g/β, α/g〉+ w ·R〈ζ/βt, ε/ζ〉[g] .

Since ζ/βt, ε/ζ and g generate R〈ζ/βt, ε/ζ〉[g] this means

R〈ζ/βt, ε/ζ〉[g] = R〈g/β, α/g〉+ w ·R〈ζ/βt, ε/ζ〉[g] . (2.14)

Applying (2.14) to itself n-times yields

R〈ζ/βt, ε/ζ〉[g] = R〈g/β, α/g〉+ wn ·R〈ζ/βt, ε/ζ〉[g]

And thus, since |w| < 1 and R〈g/β, α/g〉 is complete, we get

R〈ζ/βt, ε/ζ〉[g] = R〈g/β, α/g〉 ,

which proves the theorem in this special case.In the general case, use an etale base change on A so that Aet is an etale

neighborhood of x. Let X et := X ×A Aet and X ′et := X ′ ×A Aet be the corre-

sponding varieties for X. We can chose the etale base change in such way thatX et consists of n components which decompose in X ′et in n disjoint compo-nents, where n is the number of points in the normalization X ′ over x. In thiscase X et \ x also decomposes into n disjoint components.

Let X et be a lifting of X et. In particular we have X et+ (x) →X+(x) by Corol-

lary 2.2.11. Since X et \ x consists of n disjoint components, X1 := x ∈X et ; |f(x)| = 1 consists of n disjoint components as well. Now look atXε0 := x ∈ X et ; |f(x)| ≥ ε0 and consider a function h ∈ OX(X1) whichbehaves like a constant on any connected component with pairwise differentabsolute values unequal to zero on the components of X1. Since the image ofOX(Xε0) is dense in OX(X1), we can choose a function h ∈ OX(Xε0) whichapproximates h such that |h − h|X1 | < min|h(x)| ; x ∈ X1. The function|h| : Xε0 → R is continuous with regard to the G-topology on Xε0 . Since theabsolute values on the components of X1 are different, we can choose a δ for

26

2.5. Formal blow-ups

i 6= j so that ||h|i − |h|j | > 2δ > 0, where |·|i is the absolute value on the

i-th component of X1. Since |h| is continuous, there is an ε0 so that for every

x ∈ Xε0 we get an y ∈ X1 so that∣∣∣|h(x)| − |h(y)|

∣∣∣ < δ. By our choice of δ this

is only possible if Xε0 also consists of n disjoint components.Now look at the i-th component Xi of X et. Similar to the special case, choose

gi ∈ OX′(X′i) as a local parameter of x′i. We can extend this function to X ′et

by zero on the other components. Now lift gi to gi on X1. As in the special casewe see that, for εi > ε0, we can chose this lift in OX(Xεi). As above, gi will,after adjusting εi, generate the algebra of x ∈ Xi ; ε ≤ |f(x)| < 1. We get, asin the special case, that the i-th component Xi of Xε0 will satisfy the theorem,e.g. that x ∈ Xi ; εi ≤ |f(x)| < 1 is isomorphic to a semi-open annulus withlocal parameter gi. Since gi vanishes on all the other components, choosing anε ≥ maxi=0,...,n εi, we get that x ∈ X et ; ε ≤ |f(x)| < 1 is isomorphic to adisjoint union of semi-open annuli.

Since the base change is isomorphic on the formal fiber the theorem is alsoproved in the general case.

For the additional assertion of the theorem, let h ∈ OX(X+(x)) satisfy τ =ordx′i0

(h) < ∞ for some index i0. Since the base change does not change the

fact that gi0 is a local parameter of the reduction of the annulus Ri0 we havethe equation

h = u · gτi0 .

By lifting this equation we see that, up to a unit, h|Ri0 equals gτi0 , the pullbackof the coordinate of Ri0 .

2.5. Formal blow-ups

In this section we want to define the concept of a formal blow-up.

Definition 2.5.1. Let X be an analytic curve with reduction π : X → Xcorresponding to the admissible formal covering U = (Ui)i∈I . Let f ∈ OX(Ui)be a function such that f has an isolated zero x in Ui so that x 6∈ Uj for anyj 6= i and ε ∈ K with |ε| < 1. The formal blow-up corresponding to f and ε isthe admissible formal covering U \ Ui ∪ Ui(|f | ≤ ε), Ui(|f | ≥ ε).

Proposition 2.5.2. Let X be an analytic curve with formal covering U. LetU denote the formal blow up of U according to Ui, f and ε ∈ |K×| as defined

above. Then U is formal. The formal blow up induces a map ϕ : ˆX → X whichis an isomorphism on X \ x and is the identity on X.

Proof. The covering U is admissible because Ui(|f | ≤ ε) and Ui(|f | ≥ ε) arerational subdomains and as such admissible open. We have |f | = ε on U ′i :=Ui(|f | ≤ ε). This means that f/ε is in Ui(|f | ≤ ε) and the reduction of Ui(|f | =ε) equals the set U ′

i,f/ε. Similarly Ui(|f | = ε) reduces to U ′′

i,ε/fon U ′′i := Ui(|f | ≥

ε).The identity on X will map every point of Ui(|f | ≤ ε) to the point x. On

Ui(|f | ≥ ε) every lift y with |f(y)| < ε of a point y is in the formal fiber of xand as such y will be mapped on x. Every other point is left unchanged.

27


Proposition 2.5.3. Let X be an analytic curve with formal covering U. Letx ∈ X be a point in the associated reduction. There is a formal covering Uwhich induces the same reduction X such that x ∈ Ui for exactly one i.

Proof. For every Ui with x ∈ Ui we can find an fi ∈ OX(Ui) which does not

vanish except in isolated points and that these points lie in another set Uj aswell and fi(x) = 0. Moreover we can assume that fi and fj have only x ascommon zero on Ui ∩ Uj . Replacing Ui for all but one affected index withUi(|fi| = 1) where fi is any lift of fi does not change the reduction associatedto U.

Definition 2.5.4. Let X be a proper analytic curve with fixed reduction Xand let x ∈ X be a point. Let f and ε < 1 be as in Theorem 2.4.1 so thatx ∈ X+(x) ; ε ≤ |f(x)| < 1 consists of n disjoint semi-open annuli. Wedefine the proper curve X x by pasting n discs B1 into the formal fiber X+(x)via these annuli.

To be more precise, let

ϕi : Ri −→z ∈ B1 ; ε1/ti ≤ |ζ(z)| < 1

be the maps of Theorem 2.4.1 where Ri is the ith connected component ofx ∈ X ; ε ≤ |f(x)| < 1 and let Bi = B1 together with

ψi : Bi(|x| > ε1/ti) −→z ∈ B1 ; ε1/ti ≤ |ζ(z)| < 1 ; x 7−→ ε1/ti/x

Then X x is defined by X+(x), B1, . . . , Bn and the gluing relations ψ−1i ϕi.

Proposition 2.5.5. There is an canonical admissible formal covering U =(Ui)

ni=0 of X x. The associated reduction X x consists of n components which

are rational curves joined by a single singular point x with the same type ofsingularity as x in X.

Proof. By [BL85, Prop. 4.1.] the covering consisting of U0 = X x\(B+1 ∪· · ·∪B+

n )and Ui = Bi and B+

i as the open unit ball is a formal covering. The associatedreduction has n affine lines derived from the Bi glued together with a singlepoint x with X+(x) = X x

+(x). So the type of singularity is the same accordingto Proposition 2.2.6.

Definition 2.5.6. We define g(x) = g(X x) to be the genus of X x. We furtherset n(x) as the number of connected components of X x \X+(x).

Lemma 2.5.7. If g(x) = 0 and n(x) = 1 then x is a regular point. If g(x) = 0and n(x) = 2 then x is an ordinary double point.

Proof. If g(X x) = 0 we know that X x is a rational curve and as such isomorphicto P1. Therefore we know that X+(x) is isomorphic to P1 \ B1 if n(x) = 1 andP1 \ (B1∪B1) if n(x) = 2. In the first case we get an open disc, in the secondan open annulus, proving our assertion by Proposition 2.3.1 and Proposition2.3.2.

28

2.6. The stable reduction theorem

Definition 2.5.8. Let X be a projective connected curve over k consisting ofn components. The cyclomatic number z(X) is the number

z(X) =∑x∈X

(n(x)− 1)− n+ 1

where n(x) is again the number of points lying over x in the normalization.

Note. Since an ordinary double point has n(x) = 2, this definition coincideswith the one we give in Chapter 4 where the cyclomatic number is defined asthe number of double points minus the number of components plus one.

The main missing part of the semi-stable reduction theorem is the genusformula. We do not give a proof for this, and just state the results here.

Theorem 2.5.9. We get

g(X) =n∑i=1

g(X ′i) +∑x∈X

g(x) + z(X)

Proof. [BL85, Section 4]

With this formula, one can relate the genus of X x with the genus of pointsgenerated in a formal blow-up. If one chooses the function f correctly, one getthe following lemma, which guarantees, that the singularities will always getbetter by a formal blow-up.

Lemma 2.5.10. Let K be an algebraically closed non-Archimedean field. LetX be a projective analytic curve over K with g(X) ≥ 1. There is a mero-morphic function f and an ε ∈ |K×| such that the formal covering given byx ∈ X ; |f(x)| ≤ ε and x ∈ X ; |f(x)| ≥ ε fulfills g(x) < g(X) for allx ∈ X.

Proof. See [BL85, Lemma 7.2]

2.6. The stable reduction theorem

Definition 2.6.1. A projective, connected, reduced curve X/k that has onlyordinary double points as singularities is called semi-stable. A semi-stable curveX/k is stable if moreover every component that is isomorphic to P1

k meets therest of X in at least three points and the arithmetic genus of X is at least two.

Theorem 2.6.2. Let X be a projective analytic curve with reduction X over a

field K which is algebraically closed and complete. Then there is a reduction ˆXgenerated from X by a finite amount of formal blow ups, that has semi-stablereduction.

Proof. Since X+(x) = X x+(x) we can apply 2.5.10 to every point of X which is

not regular or an ordinary n-fold point. By Induction it follows that every pointis an ordinary n-fold point after a finite amount of formal blow-ups. Ordinaryn-fold points can be broken down further by the methods of [BL85, Chapter 5]to get to ordinary double points which proves the assertion.

29


Proposition 2.6.3. Let X be a projective rigid analytic curve with reductionX. Let X1, . . . , Xk be a selection of irreducible components of X such that⋃ki=1 Xi 6= X. There is a formal covering U of X with associated reduction X ′

and a finite number of point x1, . . . , xk such that X\⋃ki=1 Xi = X ′\x1, . . . , xk.

Proof. According to Riemann–Roch there is a rational function f such thatf(x) = 0 if and only if x ∈

⋃ki=1 Xi which is not constant on every other

irreducible component. A meromorphic lift f of f gives the formal coveringconsisting of U1 = x ∈ X ; |f(x)| ≤ 1 and U2 = x ∈ X ; |f(x)| ≥ 1. Thesesets are affinoid according to Proposition 1.10.4 and the covering is formal sincethe intersection x ∈ X ; |f(x)| = 1 is an affinoid subvariety of both U1 andU2.

Take a point x ∈ Xi with i > k i.e. f is not constant on Xi. We can assumewithout restriction that f(x) = c for c ∈ K. The formal fiber of x can bewritten as X+(x) = x ∈ U ; |f(x) − c| < 1 where U is the preimage of anaffine neighborhood U of x such that x is the only point in U with f(x) = c. Butx ∈ U ; |f(x)− c| < 1 is a formal fiber in x ∈ X ; |f(x)| = 1 as well, so theidentity on X reduces to an isomorphism on these components. All the pointsof⋃ki=1 Xi will be mapped on a connected component of x ∈ X ; |f(x)| < 1

which is a finite disjoint union of formal fibers in x ∈ X ; |f(x)| ≤ 1. So thecomponents are mapped to a finite set of isolated points in X ′.

Definition 2.6.4. The reduction X ′ of Proposition 2.6.3 is called the blow-down of the components X1, . . . , Xk in X.

Theorem 2.6.5. Let X be a projective analytic curve of genus at least two andwith semi-stable reduction X, associated to the formal covering U. There is aformal covering U of X inducing stable reduction.

Proof. Let Xi be a rational component of X which intersects the rest of thecurve in only two ordinary double points. Since the genus of X is at leasttwo, we know that Xi is not the only irreducible component of X. This meansXi\Sing X is isomorphic to P1 missing two discs. But then Xi := π−1(Xi\X) isisomorphic to K〈ζ, ζ−1〉 with ζ defined on the formal fiber of the double points.In fact ζ is a coordinate of both these open annuli.

We blow down this component using Proposition 2.6.3. This maps the wholecomponent to a single point x ∈ X ′. The formal fiber of this point is the unionof SpK〈ζ, ζ−1〉 with two open annuli, each of which having ζ as coordinate.This results in an open annulus, again with ζ as coordinate, so x is an ordinarydouble point according to Proposition 2.3.2. Therefore X ′ is again semi-stable,but misses the component Xi of X. Repeating this process until all rationalcomponents with only two intersections with the rest of the curve are goneyields a stable reduction.

30

2.7. Examples

2.7. Examples

Let us show the methods of this chapter in the example of hyperelliptic curves.Let X be defined by

U1 := SpK〈ξ, η〉/(η2 − ξ(ξ − λ2) . . . (ξ − λn))

U2 := SpK〈σ, τ〉/(τ2 − σ(1− σ)(1− λ1σ) . . . (1− λnσ))

with n = 2g − 1 and gluing relations ξ = 1/σ and η = τ/σn+1. We can assumewithout restriction that |λi| ≤ 1 by application of an affine map. This coveringis formal with associated reduction X given by

U1 := Spec k[X,Y ]/(Y 2 −X(X − 1)(X − λ1) . . . (X − λn))

U2 := Spec k[S, T ]/(T 2 − S(1− S)(1− λ1S) . . . (1− λnS))

and corresponding gluing relations. The curve X is smooth if λi 6= λj for i 6= jor equivalently if |λi − λj | = 1 for i 6= j.

Otherwise we can move a singularity to the point zero so that we can assumewithout restriction that the function f = X has a single isolated zero in theoffending singularity on U1. We choose f = ξ as a lift. Assume that the λi aresorted by absolute value and that |λi| < 1 for i ≤ k and |λi| = 1 for i > k. Set

ε = max1<i≤k

|λi| .

There is no branching point in x ∈ U1 ; ε < |ξ(x)| < 1. So this set is com-posed of one annulus if k is odd and two disjoint annuli if k is even with acoordinate calculated by the covering morphism η. Blowing U1 up with param-eters f and ε gives

U ′1 := SpK〈ξ, η, ζ〉/(ξ − εζ, η2 − ξ(ξ − λ2) . . . (ξ − λn))

U ′′1 := SpK〈ξ, η, ζ〉/(ζξ − ε, η2 − ξ(ξ − λ2) . . . (ξ − λn))

After restructuring the ideals we get

U ′1 := SpK〈ξ, η′, ζ〉/(ξ − εζ, η′2 − ζ(ζ − λ2ε

−1) . . . (ζ − λkε−1)·(ξ − λk+1) . . . (ξ − λn))

U ′′1 := SpK〈ξ, η′′, ζ〉/(ζξ − ε, η′′2 − ζj(1− λ2ε

−1ζ) . . . (1− λkε−1ζ)·(ξ − λk+1) . . . (ξ − λn))

where j = 0 if k is even and j = 1 otherwise. The new variable η′ is just ηmodified by a constant and η′′ is η/ξdk/2e also modified by a constant. Thisgives the reductions

U ′1 := Spec k[Y ′, Z]/(Y ′2 − Z(Z − λ2ε−1) . . . (Z − λkε−1)

)U ′′1 := Spec k[X,Y ′′, Z]/

(XZ, Y ′′2 − Zj(1− λ2ε−1Z) . . . (1− λkε−1Z)·

(X − λk+1) . . . (X − λn))

31


We see that U ′′1 consists of two irreducible components, both hyperelliptic curvesjoint together in a single double point if j = 1 or two ordinary double points ifj = 0. The variety U ′1 just closes the new curve.

The remaining singularities can be handled by the same process. Thereforethe stable reduction of a hyperelliptic curve consists of hyperelliptic curvesjoined together by ordinary double points.

32

3. Group objects and Jacobians

In this chapter we want to give an overview of the theory of group objects andJacobian varieties, which we will need in the next chapter.

3.1. Some definitions from category theory

Definition 3.1.1. A locally small category C is a category in which Hom(X,Y )is a set for any object X,Y ∈ C.

Definition 3.1.2. An object I ∈ C is called initial if the set Hom(I,X) containsexactly one element for any object X ∈ C. Dually an object T is called terminalif Hom(X,T ) contains exactly one element. An object that is both initial andterminal is called zero object of C

Proposition 3.1.3. Initial, terminal and zero objects of a category are uniqueup to isomorphism.

The unique morphism in Hom(I,X) and Hom(X,T ) is denoted by 0. If C hasa zero object then 0 ∈ Hom(X,Y ) denotes the unique morphism that factorsthrough the zero object.

Definition 3.1.4. Let C be a category with a zero object. For any morphismϕ ∈ Hom(X,Y ) we call k ∈ Hom(K,X) the kernel of ϕ if ϕ k = 0 and if thereis for any t ∈ Hom(T,X) with ϕ t = 0 a unique morphism u ∈ Hom(T,K)such that t = k u. In other words we have the universal property

Kk //

0

&&X

ϕ // Y

T

u

OOt

>>0

77

Dually we call c ∈ Hom(Y,C) the cokernel of ϕ if the following universalproperty

Xϕ //

0

''

0

&&Y

t

c // C

uT

holds.

33


Definition 3.1.5. Let X,Y, S ∈ C be objects with morphisms ϕ ∈ Hom(X,S)and ψ ∈ Hom(Y, S). The pullback or fibered product of ϕ and ψ is an objectP ∈ C together with morphisms p1 ∈ Hom(P,X), p2 ∈ Hom(P, Y ) such thatthe following universal property

T

ut1

t2

Pp1

~~

p2

X

ϕ

Yψ

S

holds.Dually an object P with morphisms i1 ∈ Hom(X,P ), i2 ∈ Hom(Y, P ) is called

the pushout of ϕ ∈ Hom(S,X), ψ ∈ Hom(S, Y ) if it adheres to the universalproperty

T

P

u

OO

X

i1

>>t1

AA

Y

i2

``t2

]]

S

ϕ``

ψ>>

.

If ϕ and ψ are clear from the context we write X ×S Y for the pullback andX∐S Y for the pushout. The pullback relative to the terminal object is called

the product of X and Y and the pushout relative to the initial object is calledthe coproduct of X and Y . If t1 and t2 are the morphisms of the diagram wewrite t1 × t2 : T → X ×S Y or (t1, t2) : X

∐S Y → T for the respective unique

morphism of the universal property.If x : T → X and y : T → Y as well as f : X × Y → Z are morphisms we

write f(x, y) for the morphism f (x× y).

Definition 3.1.6. A locally small category C is called additive if it has a zeroobject, all finite products exist and Hom(X,Y ) has the structure of an abeliangroup, compatible with concatenation of arrows.

Proposition 3.1.7. An additive category has all finite coproducts and the co-product X

∐Y is canonically isomorphic to X × Y for all objects X and Y .

Proof. We only need to show that X × Y is the coproduct of X and Y . We seti1 = idX ×0 and i2 = 0× idY . For any t1 : X → T, t2 : Y → T we set

u = p1 t1 + p2 t2 : X × Y → T .

34

3.2. Group objects

We then have

ij u = ij (p1 t1 + p2 t2)

= ij p1 t1 + ij p2 t2= tj

for j = 1, 2. Note that one needs the commutativity for u to be well definedand unique.

Definition 3.1.8. An additive category C is called pre-abelian if it containsevery kernel and cokernel.

Definition 3.1.9. A pre-abelian category is called abelian if every monomor-phism is the kernel and every epimorphism is the cokernel of some morphism.

3.2. Group objects

Let C be a locally small category with a terminal object 0 and all pullbacks.Let us also write 0 for the terminal morphism and pi for the ith projection ofthe product. Likewise, we will omit if no confusion is possible.

Definition 3.2.1. Let C be a category as above. We write PC for the categorywith the objects (X, e : 0→ X) where X is an object in C and e is a section of0 and the morphisms of C which respect e, i.e.

HomPC(X,Y ) = ϕ ∈ HomC(X,Y ) ; eY = ϕ eX .

Remark. The category PC has 0 as the zero object and every morphism f : X →Y has the kernel p1 : X ×Y 0→ X as the universal property of the pullback isjust the universal property of this kernel.

Definition 3.2.2. A group object G in C is an object G together with threemorphisms e : 0→ G,m : G×G→ G, i : G→ G such that

1. m((mp12)×p3) = (m(p1×(mp23)))α with respect to the canonicalisomorphism α : (G×G)×G →G× (G×G),

2. m (idG×(e 0)) = m ((e 0)× idG) = idG,

3. m (idG×i) = m (i× idG) = e 0.

A group object is called commutative if moreover

4. m = m σ holds for σ := p2 × p1 : G×G→ G×G.

Note. To clarify notation, we will use f + g : T → G or f · g for m(f, g) wheref, g : T → G are morphisms. Likewise, we use −f or f−1 for i f and 0 or 1 forthe morphism e 0. When dealing with two different group objects, we will useadditive notation for one and multiplicative notation for the other to furtherclarify where the operation takes place. If the group object is commutative weuse fractions for a compact notation of the inverse of the multiplicative notation.

35


Note that groups are just the group objects of the category of sets, which wecall abstract groups. Many elementary properties of abstract groups carry overto group objects. Since e is a section of 0, we know that 0 is epic and can usethe usual proofs to show that e and i are uniquely determined by the morphismm.

Group objects of a category together with the morphisms compatible withm, i and e form a category themselves which we denote by GC . We call thesemorphisms (group)homomorphisms and will specify the category if confusionis possible. As in the case of abstract groups the compatibility with m inducesthose for e and i and we get

HomGC(X,Y ) = ϕ ∈ HomC(X,Y ) ; mY (ϕ,ϕ) = ϕ mX

One checks easily that this category has 0 as zero object and the pullbacksA×C B has canonical morphisms m := mA(p11, p21)×mB(p12, p22), inducing iand e making it the pullback in GC .

Definition 3.2.3. We have a forgetful functor GC → Grp which associates thegroup of abstract points Hom(0, G) to a group object G. For any S ∈ C we canequivalently define a forgetful functor which associates the group Hom(S,G) tothe S-valued points G(S) of G.

The commutative group objects form a subcategory of GC which we write asZC . In this subcategory we see that A × B forms a coproduct of the groupobjects A and B. For any object T and a commutative group object G in Cwe can induce an abelian group structure on Hom(T,G), so ZC is an additivecategory. Both GC and ZC are a subcategory of PC .

Lemma 3.2.4. Every morphism ϕ : X → Y in GC has a kernel and this kernelis equal to the kernel of ϕ in PC.

Proof. As said above X ×Y 0 has a group structure.

In many cases the category ZC will have cokernels. Thereby ZC is a pre-abelian category.

Definition 3.2.5. Assume that all cokernels in ZC exist. For a homomorphismϕ we define the image of a morphism as imϕ := ker cokerϕ and dually thecoimage as coimϕ := coker kerϕ.

We say that a series of morphisms ϕi : Gi → Gi+1 forms an exact sequence ifimϕi = kerϕi+1. We say that an exact sequence is strict exact if the canonicalmorphism coimϕi → imϕi is an isomorphism for any i.

Note that the assertion of coimϕ being isomorphic to imϕ is automaticallyfulfilled in an abelian category. However, in the cases we are interested in, ZCis not abelian.

Proposition 3.2.6. Let ZC have all cokernels. For any morphism ϕ we haveker coker kerϕ = kerϕ and coker ker cokerϕ = cokerϕ.

36

3.3. Central extensions of group objects

Proof. Look at the diagram

Kk // G

c

ϕ // T

X

v

OOx

>>

C

u

OO

where k is the kernel of ϕ and c is the cokernel of k, while x is arbitrary suchthat c x = 0. Since c is the cokernel of k and ϕ k = 0 there is a unique arrowu : C → T such that ϕ = u c. So we get 0 = u 0 = u c x = ϕ x. But kis the kernel of ϕ, so there is a unique arrow v : X → K making the diagramcommutative which just means that k is the kernel of c, as required.

The other part of the assumption follows by the dual diagram.

Therefore a short sequence is strict exact if the first morphism is the kernelof the second and the second morphism is the cokernel of the first.

Definition 3.2.7. Let 0→ G→ E → B → 0 be a strict exact sequence in ZC .We call E an extension of B by G. Two extensions E and E′ are isomorphic ifthere is an isomorphism f such that the diagram

0 // G

id

// E

f

// B

id

// 0

0 // G // E′ // B // 0

(3.1)

commutes. We denote by Ext(B,G) the class of extensions of B by G up toisomorphism.


We want to classify these group extensions. For abstract commutative groupsthis is done by using the notion of the central extension and group cohomology.The same process can, with some restrictions, be applied to commutative groupobjects which we will sketch in the next theorem.

For a strict exact sequence 0 → G → E → B → 0, we use multiplicativenotation on G and additive notation on E and B. Since all group objects arecommutative, we use fractions for the inverse in multiplicative notation.

Definition 3.3.1. Let B and G be commutative group objects. We call theset of morphisms f ∈ Hom(B2, G) satisfying

f(y, z) · f(x, y + z)

f(x+ y, z) · f(x, y)= 1 (3.2)

for the projections x, y, z of B ×B ×B the cocycles Z2(B,G) of B with coeffi-cients in G. The coboundries B2(B,G) are morphisms given by

f(x, y) =g(x) · g(y)

g(x+ y)(3.3)

37


where g : B → G is an arbitrary morphism and x, y are the projections of B×B.

The second cohomology group of B with coefficients in G is given by

H2(B,G) = Z2(B,G)/B2(B,G) .

The restriction of Z2(B,G) to symmetric morphisms f : B2 → G wheref(x, y) = f(y, x) is denoted by Z2(B,G)s. We set

H2(B,G)s = Z2(B,G)s/B2(B,G) .

Theorem 3.3.2. Let C be a category as in Section 3.2 such that ZC has allcokernels. Let B and G be objects in ZC. The class of commutative extensions

0 // Gψ // E

ρ // B //s

ii 0

of B by G up to isomorphism which admit a section of ρ in the category C isin one-to-one correspondence to the group H2(B,G)s.

Proof. Take a symmetric cocylce f ∈ H2(B,G)s. We can assume withoutrestriction, that f(eB, eB) = eG because we can otherwise modify f with thecoboundry induced by g = f ∆, where ∆ is the diagonal morphism. We endowE := G×B with a group law by setting

e := eG × eBm := (g1 · g2 · f(b1, b2))× (b1 + b2)

i :=

(1

g · f(b,−b)

)× (−b)

ρ := p2

ψ := idG×eBs := eG × idB

where g1, g2, b1, b2, g, b are the projections of (G × B) × (G × B) and G × Brespectively.

Using the cocycle condition (3.2) for idB ×eB × eB gives

f(eB, eB) = f(eB, b) = eG

so e is neutral and i is an inverse. The associativity follows directly by thecocycle condition (3.2) and the commutativity is implied by f being symmetric.

One checks directly that ψ and ρ are indeed homomorphisms in ZC . LetX ∈ ZC be any object and ϕ : E → X a grouphomomorphism with ϕ ψ = 0so we get the following diagram

Gψ //

0

''

Eρ //

ϕ

B

uX

38


We set u = ϕ s. Let b1 and b2 be the projections of B ×B then

u b1 + u b2 = ϕ s b1 + ϕ s b2= ϕ (s b1 + s b2)

= ϕ (1× b1 + 1× b2)

= ϕ (f(b1, b2)× (b1 + b2)) .

We know that ϕ ψ (f(b1, b2))−1 = 0 so we get

ϕ (f(b1, b2)× (b1 + b2)) = ϕ (1× (b1 + b2)) = ϕ s (b1 + b2)) = u(b1 + b2) ,

which means that u is a homomorphism, which is automatically unique since ρis epic. By the same argument we get

ϕ = ϕ(p1, ρ) = ϕ(1, ρ) = ϕ s ρ = u ρ .

We also have

ker ρ = E ×B 0 = (G×B)×B 0 ,

which is isomorphic to G via ψ by the universal properties of product andpullback in C. Since the kernels in PC and ZC agree, we know that ψ = ker ρ.So ψ and ρ form a strict exact sequence.

On the other hand, for a given strict exact sequence

0 // Gψ // E

ρ // B //s

ii 0

with a section s of ρ we can assume without restriction that s eB = eE , sinceotherwise we can replace s by s− s eB.

We define r′ : E → E ; r′ = idE −s ρ, so ρ r′ = eB which implies thatthere is a morphism r : E → G ; ψ r = r′ since ψ = ker ρ in PC as well. Butr′ ψ = ψ, so r is a retraction of ψ.

Thus we get an isomorphism ϕ : G×B → E by ϕ = ψg+sb and ϕ−1 = r×ρ.We set f ′ := s b1 + s b2 − s (b1 + b2). Then ρ f ′ = eB so there is f suchthat f ′ = ψ f . We can define a group law on G×B via ϕ as

m : (G×B)× (G×B)→ G×Bm = ϕ−1 mE (ϕ p1 × ϕ p2)

= ϕ−1 (ψ g1 + s b1 + ψ g2 + s b2)

= (g1 · g2 · f(b1, b2))× (b1 + b2) .

For another section s′ of ρ we can define g : B → G such that ψ g = s′ − ssince ρ (s′ − s) = eB. The cocycles corresponding to s and s′ just differ bythe coboundry induced by g. On the other hand any morphism g : B → G canbe induced this way by setting s′ = s + ψ g. By the definition of isomorphyof extensions, this shows that two extensions are isomorphic if and only if theyinduce the same cohomology class in H2(B,G)s.

39


Definition 3.3.3. Let α : G→ G′ be a homomorphism in ZC and let

1 // G

α

ψ // Eρ // B //s

ii 1

G′

be a strict exact sequence admitting a section. Then the extension α∗E ∈Ext(B,G′) corresponding to α∗f := α f , where f is the cocycle describing E,is called the pushout of α.

Proposition 3.3.4. The pushout of a homomorphism α is the categorical the-oretic pushout of α and ψ. It is the unique extension of B by G′ such that ahomomorphism A exists such that the diagram

1 // G

α

ψ // E

A

// B

id

// 1

1 // G′ // E′ // B // 1

commutes.

Proof. The pushout of α and ψ is given by the cokernel of −α×ψ. This cokernelis just p1 + α r × ρ p2. We set A := α p1 × p2 which makes the diagramcommutative. The uniqueness of E′ follows by the uniqueness of the categoricalpushout.

Definition 3.3.5. Let γ : B′ → B be a homomorphism in ZC and

B′

γ

1 // G

ψ // Eρ // B //s

ii 1

a strict exact sequence admitting a section. The extension γ∗E of B′ by Ggiven through the cocycle γ∗f := f (γ p1) × (γ p2) where f is the cocyclecorresponding to E is called the pullback of γ.

Proposition 3.3.6. The pullback of a homomorphism γ is the categorical the-oretic pullback of γ and ρ. It is the unique extension of B′ by G such that ahomomorphism Γ exists such that the diagram

1 // G

id

// E′

Γ

// B′

γ

// 1

1 // G // Eρ // B // 1

commutes.

Proof. Since the categorical pullbacks of ZC and PC coincide we get

E′ = (G×B)×B B′ →G×B′

as underlying object for the pullback in ZC . Using this canonical isomorphismto induce a group law produces the given central extension.

40

3.4. Algebraic and formal analytic groups


We are interested in group objects of the category of algebraic varieties, i.e.separated, connected and integral schemes of finite type over an algebraicallyclosed field which we call algebraic groups and the group objects of the categoryof connected, quasi-separated, quasi-paracompact rigid K-spaces, which we willcall analytic groups. We are especially interested in analytic groups with a fixedreduction π : X → X, where X is an algebraic group, on which the group lawsare given as the reduction of the group laws on X, in other words a formalgroup scheme over R.

In the rest of this section we will collect some statements about algebraicgroups and look to generalize them to their analytic counterparts.

Proposition 3.4.1. Algebraic and analytic groups are non-singular.

Proof. Let G be the group in question and τx : G → G be the left translationby a closed point x ∈ G. This induces an isomorphism of the tangent spaceof G at 0 and at x. Since we defined algebraic and analytic groups as reducedschemes this proves the assumption.

Proposition 3.4.2. Let G and H be algebraic groups and U ⊂ G a non emptyopen subvariety. Let ϕ : U → H be a morphism compatible with the group law.Then there is a unique group homomorphism ϕ : G→ H which restricts to ϕ.

Proof. Let x ∈ G be any closed point. Then U − x is isomorphic to U andopen. G is integral and therefore irreducible, so U ∩ (x − U) is non empty.Take a = x − b from that intersection and set ϕ(x) = ϕ(a) + ϕ(b). Sinceϕ is compatible with the group law the choice of a does not matter and ϕrestricts to ϕ. Furthermore for any fixed a ∈ U we can set ϕa : U + a→ H byϕa(x) = ϕ(x− a) + ϕ(a) and obtain a morphism that coincides with ϕ on thecommon intersection. So ϕ is a morphism of algebraic varieties and respectsthe group law. It is unique since rational functions are uniquely determined byan open subvariety.

Corollary 3.4.3. Let G be a formal group scheme and H be any analyticgroup. Let U ⊂ G be a non empty formal open subset of G and ϕ : U → H bea morphism which respects the group law, i.e. ϕ(u1 + u2) = ϕ(u1) + ϕ(u2) foru1, u2, u1 + u2 ∈ U . Then there is a unique group homomorphism ϕ : G → Hwhich restricts to ϕ.

Proof. Take any point x ∈ G. We’ll get again an isomorphy between U andx−U and both are formal open subsets. Since G has an irreducible reduction,we know that U ∩ (x − U) is not empty, so we can define ϕ(x) as above. Wecan again prove locally that ϕ is a morphism and obtain uniqueness since amorphism is defined by its values on any non empty formal open subset.

The last corollary suggests the definition of a formal rational function asfollows.

41


Definition 3.4.4. A formal rational function on a smooth formal connectedscheme X over R is a pair (f, U) where U is a non-empty admissible formalopen subset of X and f ∈ OX(U). Two rational functions (f, U) and (g, V ) arecalled equal if the restrictions of f and g to U ∩ V are equal.

Proposition 3.4.5. Let (f, U) be a formal rational function on X and letV ⊃ U be another admissible formal open subset of X. If there is an extensionof f on V , it is unique.

Proof. We can assume that U is a rational subdomain of an affinoid set V . Thenthe proposition follows according to the identity theorem of power series.

Remark. On should note that meromorphic functions are not equivalent torational functions as one might expect. Just take P1

K as U1 = SpK〈ζ〉 andU2 = SpK〈η〉 glued on the subset SpK〈ζ, η〉/(ζη−1). A meromorphic functionwithout poles can be written as a power series on both U1 and U2, whichimplies that it must be constant. So a meromorphic function on P1

K is alwaysthe quotient of two polynomials in ζ as one might expect. But K〈ζ〉 containspower series that are not the quotient of two polynomials, for example if |a| > 1then 1/(ζ − a) has a square root in K〈ζ〉 as simple calculations show.

Proposition 3.4.6. A short exact sequence of commutative algebraic or formalanalytic groups

0→ Gψ−→ E

ρ−→ B → 0

is strict exact if and only if the induced sequence on the tangent spaces

0→ TGdψ−→ TE

dρ−→ TB → 0

is also exact.

Proof. As in [Ser88, III, §3 Cor. 2 and 3] the condition on the tangent spacesimplies that G is a closed subvariety of E and that B is the geometric quotientE/G. For any object X in ZC we get the exact sequences

0→ Hom(B,X)→ Hom(E,X)→ Hom(G,X)

0→ Hom(X,G)→ Hom(X,E)→ Hom(X,B)

which implies that ρ is the cokernel of ψ and ψ is the kernel of ρ. So thesequence is strict exact.

Starting with a strict exact sequence we also get

0→ Hom(X,G)→ Hom(X,E)→ Hom(X,B)

for any X in ZC . But since kernels in ZC and PC are equal, the same sequenceis exact for PC . We can expand C to include non-reduced varieties. Using X =Spec k[ε] gives the desired sequence on the tangent spaces, with the surjectivityof the last morphism implied by dimension.

Remark. The condition on the tangent spaces make ψ a reduced closed immer-sion and ρ a smooth morphism.

42


We know how to describe the extensions of algebraic/analytic groups if the

strict exact sequence 0 → Gψ−→ E

ρ−→ B → 0 admits a regular section. We putH2

reg(B,G)s for these extensions.

If ρ only admits a rational section, we end up with a “birational” group Ewith a group law only defined on a non-empty open subset. There is alwaysa unique algebraic group birationally equivalent to a given birational group asseen in [Ser88, VII, §1 Prop. 4]. So we still end up with a unique extension thisway. We put H2

rat(B,G)s for this subgroup of Ext(B,G). Since the extensionis constructed by gluing techniques the same works for the formal analyticcase. Note that a rational section of a morphism of formal analytic groups isa morphism defined on a formal open subset and not necessarily meromorphic.We will construct this group explicitly for the case that G is a linear group inProposition 3.5.3.

Proposition 3.4.7. Let

0→ G→ Eρ−→ B → 0

be a strict exact sequence of formal groups schemes over R with an algebraicallyclosed field of fractions with the reduction

0→ G→ E → B → 0

of algebraic groups. Then the sequence has a rational section if and only if thereduced sequence has a rational section.

Proof. As any section of the analytic sequence reduces to a section of the re-duction, this part of the proof is trivial.

On the other hand, since we are talking about rational sections, we canassume E and B to be affinoid. Then the proof is given by Proposition 1.9.7.

Proposition 3.4.8. Let 1 → G → Eρ−→ B → 1 be a strict exact sequence of

connected commutative algebraic groups, where G is linear. Then the sequencehas a rational section.

Proof. We will sketch the proof as it is done in [Ser88, VII, §1 Prop. 6].

Let x be any point of B. Then Ex := ρ−1(x) = E×B Spec k(x) is a principalhomogeneous space for G over k(x), i.e. G acts transitively on Ex via multipli-cation on E and this action is injective for a fixed e ∈ Ex. The action and theinduced division map are both regular.

A rational section of ρ is the same as a k(x) rational point in Ex where x isthe generic point of B, in other words Ex is isomorphic to G and the trivialhomogeneous space.

By a result of Lang and Tate in [LT58, Prop. 4] the isomorphism classesof principal homogeneous spaces over an arbitrary field k are in one-to-onecorrespondence to H1(gs, G), where gs is the Galois group of ks/k with thetopology induced by etale extensions, where ks is the separable closure of kacting on G by action on the coordinates of the points.

43


The group H1(gs, G) is trivial for a commutative linear group G. We will onlyshow this for the case G = Gm,K since this is the case we are most interested in.The general case can be proven by showing the result for Ga,K and combiningthese two cases to form the arbitrary linear group G as seen in [Ser88, VII, §1Prop. 6].

So for the case G = Gm,K take a finite extension K/k of k with Galois groupA. A cocycle in H1(A,Gm,K) is a map f : A → Gm,K satisfying f(σ τ) =f(σ) · σ(f(τ)) and the coboundry are the maps bα : A→ Gm,K σ 7→ σ(α)/α.

Now for a fixed automorphism σ we can calculate the norm of f(σ) in theextension induced by 〈σ〉 by the cocycle condition to be

NKK〈σ〉(f(σ)) =

r−1∏i=0

σi(f(σ)) =r−1∏i=0

f(σi+1)

f(σi)= 1 .

So according to Hilbert 90 there is an α ∈ K with f(σ) = σ(α)/α i.e. f is acoboundry.

So since there is only one principal homogeneous space of G over k(x) weconclude that Ex is isomorphic to G and therefore has a k(x)-rational pointleading to the needed section.

3.5. Extensions by tori

There is a different way to interpret H2rat(B,G)s. A rational section of B to E

can be translated by the multiplication on B and yields a family of (admissible)open sets Ui on B, each with a section si. Together with the morphism ψ : G→E of the exact sequence this gives open embeddings sip1×ψp2 : Ui×G→ E,making E a fiber space of B with the fiber G. So if GB is the sheaf of germs ofregular functions from B to G, we see that there is a map

π : H2rat(B,G)s → H1(B,GB) .

To make things more explicit let f correspond to the extension E with therational section s : U → E where U is an open subset of B. Recall that then

ψ f(b1, b2) = s(b1) + s(b2)− s(b1 + b2)

for any b1, b2 ∈ U with b1 + b2 ∈ U .

The section s gives an isomorphism ϕ0 between a (formal) open subset of Eand U ×G. Covering B with the sets b+ U for b ∈ U let us calculate

sb : b+ U → B ; b+ u 7→ s(b) + s(u)

as the translated sections with ϕb as corresponding isomorphisms.

So for b1, b2 ∈ U we get for

ϕ−1b1 ϕb2 : [(b2 + U) ∩ (b1 + U)]×G→ [(b1 + U) ∩ (b2 + U)]×G

44


the map defined by

ϕ−1b1 ϕb2(u, g) = ϕ−1

b1(ψ(g) + s(b2) + s(u− b2))

= ϕ−1b1

(ψ(g) + s(b1) + s(u− b1)

− s(b1)− s(u− b1) + s(u)− s(u) + s(b2) + s(u− b2))

= ϕ−1b1

(ψ(g) + sb1(u)− ψ f(b1, u− b1) + ψ f(b2, u− b2))

=

(u, g · f(u− b2, b2)

f(u− b1, b1)

)So we see that

π(f) =

(f(u− b2, b2)

f(u− b1, b1)

)bi∈U

is the image of a cocycle.

Proposition 3.5.1. Let 0 → G → E → B → 0 be a strict exact sequence offormal analytic or algebraic groups. The map π : H2

rat(B,G)s → H1(C,GB) isa group homomorphism which commutes with pullbacks and pushouts.

Proof. As already noted we get π(f) = (f(u − b2, b2)/f(u − b1, b1))bi∈U as theimage of a cocycle. So we see directly that π(f · g) = π(f) · π(g) after choosinga common open set U so π is a group homomorphism.

With that we can directly calculate for any group homomorphism α : G→ G′

that

π(α∗f)b1,b2 =α f(u− b2, b2)

α f(u− b1, b1)

= α (f(u− b2, b2)/f(u− b1, b1))

= α∗π(f)b1,b2

holds and that for any group homomorphism γ : B′ → B

π(γ∗f)γ(b1),γ(b2) =f(γ(u− b2), γ(b2))

f(γ(u− b1), γ(b1))

=f(γ(u)− γ(b2), γ(b2))

f(γ(u)− γ(b1), γ(b1))

= γ∗π(f)γ(b1),γ(b2)

holds as well, which concludes the proof.

Proposition 3.5.2. Let G be a commutative linear and B be a commutativeprojective algebraic or formal analytic group. Then the map π : H2

rat(B,G)s →H1(B,GB) is injective.

Proof. Let again be E the extension corresponding to f ∈ H2rat(B,G)s and

s : U → E the rational section of ρ, defined on the open subset U of B. If fis in kerπ then f(x − b, b) = s(x − b) + s(b) − s(x) is regular on all of U × Uafter the change by a coboundry. But then s is a regular section which in turnimplies that f is regular on B ×B and since B is projective and G is affine, fis necessarily constant and thus induced by the constant coboundry.

45


Proposition 3.5.3. Under the assumptions of Proposition 3.5.2 the image ofthe map π : H2

rat(B,G)s → H1(B,GB) consists of all elements x ∈ H1(B,GB)for which m∗Bx = p∗1x · p∗2x holds on B ×B.

Proof. First we note that for any f ∈ H2rat(B,G)s we get

m∗Bπ(f) = π(m∗Bf) = π(p∗1f · p∗2f) = p∗1π(f) · p∗2π(f)

according to Lemma 3.5.1 and the coboundaries of H2(B ×B,G).

On the other hand, if (Ui)i∈I is an open covering for which (gij) representsan element in H1(B,GB) we can reformulate the stated property, as

gij(x+ y)

gij(x)gij(y)=fj(x, y)

fi(x, y)

where (fi) is in Z0(B × B,GB×B). We want to show that any fi represents apreimage of (gij).

By the definition of H2rat(B,G)s, all of the (fi) differ only by a coboundry.

For any i the fi also fulfills the cocycle condition, since

h(x, y, z) :=fi(y, z)fi(x, y + z)

fi(x+ y, z)fi(x, y)

defines the same function for all i ∈ I and is thus constant as a regular functionfrom the projective variety B×B×B to the affine variety G and h(0, 0, 0) = 1.

This map is obviously a group homomorphism. It is also injective. Supposewe have (gij) in the kernel i.e.

gij(x+ y)

gij(x)gij(y)=hj(x+ y)hi(x)hi(y)

hi(x+ y)hj(x)hj(y).

Changing (gij) by the coboundry (hj/hi) gives

gij(x+ y) = gij(x) · gij(y) .

So gij is a group homomorphism from B to G by Proposition 3.4.2. Thereforegij is constant as a map from a projective to an affine space and (gij) is thetrivial bundle.

For f ∈ H2rat(B,G)s this trivialization for π(f) is given by

π(f)bi,bj (x+ y)

π(f)bi,bj (x) · π(f)bi,bj (y)=f(x+ y − bj , bj) · f(x− bi, bi) · f(y − bi, bi)f(x+ y − bi, bi) · f(x− bj , bj) · f(y − bj , bj)

=f(x, y) · f(x− bi, y) · f(y − bi, bi)f(x− bj , y) · f(y − bj , bj) · f(x, y)

=f(x− bi, y − bi) · f(x+ y − 2bi, bi)

f(x− bj , y − bj) · f(x+ y − 2bj , bj)

which for bi = 0 recovers the cocycle f .

46


Remark. One can even construct an explicit extension of B by G this way. Takea principal fiber space L of x. Then m∗Bx = p∗1x+ p∗2x gives a regular functiong : L × L → L compatible with the multiplication on B. One can use thisfunction g to define a group law on L, see [Ser88, pg. 182 et seq.] for details.Since the construction works as well for analytic groups, this shows that allrational cocycles indeed come from an extension.

Proposition 3.5.4. Let B be an abelian variety or the generic fiber of a properformal group scheme. Then an extension of B by Gm,K is equal to a point ofPicτ (B), the dual abelian variety of B.

Proof. We have already seen in Proposition 3.5.2 that the morphism

π : H2rat(B,Gm,K)s → H1(B,O×B)

is injective and that its image consists of the line bundles L ∈ H1(B,O×B) whichfulfill m∗BL = p∗1L⊗ p∗2L which by the Theorem of the square as in [Mum08, II.8./III. 13.] happens if and only if L ∈ Picτ (B).

Proposition 3.5.5. Let B be an abelian variety or the generic fiber of a properformal group scheme and t ∈ N. Then an extension of B by Gt

m,K is equal to a

point of Picτ (B)t.

Proof. This follows from the previous proposition and the fact that the coho-mology group H1(B,O×B

t) equals H1(B,O×B)t by definition.

We can formulate the last proposition without introducing coordinates.

Corollary 3.5.6. Let B be a proper formal group scheme. Then an exten-sion of the generic fiber of B by a torus T is equal to a group homomorphismHom(T,Gm,K)→ Picτ (B) of the character group of T to the translation invari-ant line bundles over B.

Proof. We can choose coordinates on T so that we get T →Gtm,K . An extension

1→ Gtm,K → E → B → 1 gives rise to the map

ϕ : Hom(Gtm,K ,Gm,K)→ Picτ (B)

via α 7→ α∗E which is a total fiber space of a line bundle in Picτ (B). By thedefinition of the pushout, this is a group homomorphism. On the other handany such group homomorphism is determined by the images of the projectionswhich give the point of Picτ (B)t of Proposition 3.5.5.

Corollary 3.5.7. Let 0→ T → E → B → 0 be an extension of the generic fiberof a proper formal group scheme by a torus T . An S-valued point σ : S → Eis equivalent to a family of S valued points σχ : S → PB×φ(χ) with σχ1+χ2 =σχ1 ⊗ σχ2 and where φ is the homomorphism of the last corollary and PB×B′ isthe Poincare bundle over B.

In other words we can write E =∏ni=1 PB×φ(e′i)

where e′i is a basis of thecharacter group and a point t = (t1, . . . , tn) ∈ E corresponds to the family

tχ = t⊗m′11 ⊗ · · · ⊗ t⊗m

′n

n ∈ PB×φ(χ) where χ = m′1e′1 + · · ·+m′ne

′n with m′i ∈ Z.

47


Note. If one does not care about the embedding ψ : Gm,K → E of Gm,K inthe extension E, one needs to divide by the automorphism group of Gm,K .Therefore two line bundles L and L−1 yield the same extension up to ψ.

Definition 3.5.8. Let 1 → Gtm,K → E → B → 1 be an extension of formal

group schemes. Denote by 1→ Gtm,K → E → B the pushout given by the map

Gtm,K → Gt

m,K on the generic fibers. We define |·| : E → R+0t

by x 7→ |a| with

a ∈ Gtm,K and x · a−1 ∈ E. A lattice M in E is an analytic subgroup of E so

that |·||M is injective and − log|M | is a lattice in Rt. The rank of M is just therank of − log|M |.

Remark. There is a more formal approach to define a value on E. We knowthat E is a line bundle over the formal scheme B. So the defining cocylces (gij)have absolute value 1 as they are functions of a formal scheme to Gm,K . Thisinduces the absolute value on E with the absolute value of Gm,K and leads tothe same value that we defined.

Proposition 3.5.9. Let E be an extension of the generic fiber of a properformal group scheme B by a Torus T as before. Let M be a lattice of full rankin E. Then E/M is again a proper analytic group variety.

Proof. We writeM

0 // T

ψ // Eρ //

B // 0

A

with A = E/M . As an abstract group A obviously has a group structure. Eis a total fiber space of a line bundle, so we can find an affinoid subset U ⊂ Bsuch that E can be covered by a finite amount of translations of ρ−1(U) andρ−1(U) = Gt

m,K × U . We write Tc for the annulus T 〈c−1ζ, cζ−1〉 where |c| < 1

and ζ is a coordinate on T . Since the rank of the lattice is full we find c ∈ K×such that the translates of Tc × U already cover A. Since B is assumed to beproper and is formal so its generic fiber is proper in the rigid analytic sense andwe can cover U by relatively compact subsets. This means that A is proper.

Proposition 3.5.10. Let A be proper analytic group and let E be an openformal analytic subgroup of A. Suppose there is a morphism ψ : Gt

m,K → A

that restricts to ψ : Gtm,K → E and gives E as extension of a proper formal

group scheme B by Gtm,K . Suppose that there is an ε ∈ R such that every

element of A can be written as ψ(g) · e with e ∈ E and ε ≤ |g| ≤ ε−1. Then Ais isomorphic to E/M where E is the pushout as in Definition 3.5.8 and M isa lattice of full rank.

Proof. We can write the elements of E as tuples (g, e) ∈ Gtm,K × E with the

equivalence relation (g, e) ∼ (g′, e′) if and only if ψ(g/g′) = e′/e according toProposition 3.3.4.

48


Define the group homomorphism η : E → A by setting η(g, e) = ψ(g) · ι(e)with ι : E → A as the open embedding of E in A. The kernel of η is a latticein E, since ι and ψ are injective and the value of an element (g, e) of E is thevalue of g.

The assumption that every element of A can be written as ψ(g) · e whereg is bounded and e ∈ E implies the surjectivity and that the lattice is of fullrank.

49

4. The Jacobian of a formal analyticcurve

4.1. The cohomology of graphs

In this section G(V,E) will be an arbitrary connected graph with vertex set Vand edge set E.

Definition 4.1.1. A graph G(V ′, E′) with V ′ ⊂ V and E′ ⊂ E, where everyedge of E′ has source or target in V ′ is called a subgraph of G(V,E). A subgraphis called complete, if E′ contains all edges of E with source and target in V ′.

We want to discuss homology and cohomology of a connected graph. For thisto be defined, we need to fix a topology on G. This is done by declaring allcomplete subgraphs of G to be open. Since cycles in a homology group have anorientation, we will have to choose an arbitrary orientation for each edge e ∈ E.

In this section a simple cycle is a directed closed path in the graph whichvisits an edge only once. The general term cycle is used in the homological senseof the word, i.e. a chain of edges with zero boundary. Cycles can be obtainedby taking formal sums of simple cycles modulo the standard concatenation ofcycles. We say that an edge ends in a vertex if said vertex is either target orsource of the edge. The graph G may contain loops.

Definition 4.1.2. Denote by n the number vertices of G and by e the numberof edges. The number t := e− n+ 1 is called the cyclomatic number of G.

The cyclomatic number of a graph is fundamental to describe its homology,as the following proposition shows.

Proposition 4.1.3. Let t be the cyclomatic number of a connected graph G.Then one can fix t edges ε1, . . . , εt in E such that G′(V,E \ ε1, . . . , εt) is atree. Each of these edges εi corresponds uniquely to a simple cycle γi on G ina natural way.

Proof. A connected graph is a tree if and only if its cyclomatic number t equals0. If a connected graph G contains cycles, i.e. t ≥ 1 we remove an edge ε1

of one of the cycles, thereby getting a graph G with cyclomatic number t − 1.Repeating that t times yields a tree and t edges ε1, . . . , εt. The graph G(V,E \ε1, . . . , εi−1, εi+1, . . . , εt) has exactly one simple cycle matching orientationwith εi which we denote by γi. In other words this is the unique cycle in Gwhich passes exactly one time through εi but none of the εj for j 6= i.

To compute the homology groups of G we first note that the homology groupsfor the constant sheaf Z defined by our topology coincide with the homology

51

4. The Jacobian of a formal analytic curve

groups of G if we regard G as a simplicial complex of dimension 1. This givesus the group H1(G,Z) as the group of formal sums of the closed paths of Gwith coefficients in Z, i.e. the cycles of G.

Proposition 4.1.4. The group H1(G,Z) is isomorphic to Zt with t the cyclo-matic number of Definition 4.1.2. Higher homology groups vanish.

Proof. We define a group homomorphism from cycles in G to Zt, by countinghow often the cycle passes through the edges ε1, . . . , εt of Proposition 4.1.3,where the sign encodes the direction of the cycle in this edge. For an element(ai)

ti=1 of Zt we obtain a preimage as the cycle

∑ti=1 aiγi. Suppose two cycles

γ and γ′ have the same image. Then γ − γ′ is mapped to 0, i.e. it is equivalentto a cycle which passes through none of the εi. But G′(V,E \ ε1, . . . , εt) is atree, thus γ − γ′ is trivial and γ = γ′.

We can now compute the cohomology groups Hq(G,Z) via Cech cohomology.

Proposition 4.1.5. Let (A, ·) be any abelian group. Then the cohomology groupH0(G,A) is isomorphic to A. The group H1(G,A) arises as Hom(H1(G,Z), A)and is as such isomorphic to At. All higher cohomology groups vanish.

Proof. A graph is connected in the graph-theoretical sense if and only if it isconnected topologically. Therefore H0(G,A), the group of global section of theconstant sheaf A, is trivially isomorphic to A.

To calculate the higher cohomology groups we define an open covering. LetGe be the subgraph of G consisting of the edge e and the two vertices it joins.Then G = Ge ; e ∈ E(G) is an open covering of G, as the Ge are complete.With respect to this covering, we have

Cq(G, A) =∏

(e0,...,eq)∈E(G)q+1

A(Ge0 ∩ · · · ∩Geq)

and

dq : Cq → Cq+1 ; α 7→

(p+1∏k=0

(−1)kαi0,...,ik−1,ik+1,...,iq+1

)i0,...,ip+1

,

as in [Har77, pg. 218] for standard Cech cohomology. The elements of C0(G, A)therefore assign an element of A to each edge of E, whereas the elements ofC1(G, A) assign an element of A to each pair of edges which share a vertex ofG.

To get the group H1(G,A) = ker d1/ im d0 we need to discuss the kernel of d1

and the image of d0. As always, the elements of ker d1 are the elements of C1

which satisfy the cocycle relations αei,ej = αei,ek · αek,ej for every three edgesei, ej , ek sharing a vertex v. In particular, we have αe,e = 1 and αei,ej = α−1

ej ,ei .

To finish our description of H1(G,A), we need to look at the coboundariesim d0. By definition, these are the elements defined by αei,ej = βejβ

−1ei for a

0-cocycle (βe)e∈E .

52

4.1. The cohomology of graphs

Given any cycle γ =∑m

k=1 ek ∈ H1(G,Z), we get

m∏k=1

αek,ek+1=

m∏k=1

(βek+1

β−1ek

)= 1 , (4.1)

where e1 = em+1.

Conversely, suppose we have an element α of C1(G, A) such that the productof (4.1) is trivial for every cycle of G. We can define a 0-cocylce (βe)e∈E of G bysetting βe0 = 1 and recursively defining βei = βejα

−1ei,ej , if βej is already defined

and ei and ej share a vertex. Since the product of (4.1) is trivial this does notdepend on the path we choose from ei to e0.

This means that the elements of H1(G,A) are uniquely defined by the possiblevalues of the left hand side of (4.1). Therefore we get a dualizing form

Φ: H1(G,Z)× H1(G, A)→ A

by sending (γ, α) to this left hand side. This gives us an isomorphism

ϕ : H1(G, A) →At

by dualizing using the generators (γi) of H1(G,Z).

By the same argument the higher cohomology groups must be dual to thecorresponding homology groups. Since these vanish, the cohomology groupsvanish too.

Since any intersection of elements of G is connected or empty the sheaf A isacyclic on each of these intersections. This implies that Hq(G,A) = Hq(G, A)for all q ∈ N.

Definition 4.1.6. For an arbitrary edge e with target vertex v and an elementa ∈ A we define the weighted cocycle α(e, a) = (αei,ej )(ei,ej)∈E2 by setting

αei,ej =

a if ei = e, ej 6= e and ej ends in v

a−1 if ej = e, ei 6= e and ei ends in v

1 otherwise.

One checks easily that every weighted cocycle obeys the cocycle conditions.Recall that an edge ends in a vertex if said vertex is either target or source.

Lemma 4.1.7. The image ϕ(α(e, a)) of a weighted cocycle under the mapϕ : H1(G,A)→ At can be computed as

ϕ(α(e, a))i =

a if e ∈ γi and γi traverses e in the same direction

a−1 if e ∈ γi and γi traverses e in the other direction

1 otherwise.

where γi are the simple cycles of Proposition 4.1.4.

53


Proof. If the cycle γi does not contain e, every factor of (4.1) is trivial. Forγi =

∑mk=1 ek with er = e and α(e, a) = (αej ,ek)(ej ,ek)∈E2 , we have

ϕ(α(e, a))i =m∏k=1

αek,ek+1= αer−1,erαer,er+1 .

If γi traverses e in the direction of e we get αer−1,er = 1 and αer,er+1 = a. If onthe other hand γi traverses e in the opposite direction we have αer−1,er = a−1

and αer,er+1 = 1.

Corollary 4.1.8. Denote by e the edge e with reversed direction. Then we getα(e, a) = α(e, a−1).

Since the cycle γi corresponding to the edge εi was defined as the cyclecontaining εi and none of the εj for i 6= j, we see that

ϕ(α(εi, a)) = (1, . . . , 1, a, 1, . . . , 1) ,

the element of At with a at the ith component. This means that the weightedcocylces are dual to the simple cycles in the sense that

Φ(α(εi, a), γj) = aδij .

With this construction we can give an explicit inverse morphism of ϕ. Let(ai)

ti=1 be an element of At. Then the cocycle

(αej ,ek)(ej ,ek)∈E2 =

t∏i=1

α(εi, ai)

is a preimage of (ai).

4.2. The cohomology of curves with semi-stablereduction

We recall the definition of a semi-stable curve of Chapter 2.

Definition 4.2.1. A projective, connected, geometrically reduced curve X/kthat has only ordinary double points as singularities is called semi-stable.

Let X be a formal analytic curve with semi-stable reduction X. We writeXi, i = 1, . . . , n for the irreducible components of the reduction. The compo-nents Xi can at most have nodes as singularities. If there is a node which liesonly on one component, we know by Chapter 2 that it has a formal fiber whichis isomorphic to an open annulus. We can use a formal blow-up to subdividethis open annulus in order to replace this node by two nodes lying on differentcomponents. Since all statements in this section are trivial for a curve with goodreduction, this means that we can assume that the reduction X has at least twoirreducible components and that every node lies on two different components.

54

4.2. The cohomology of curves with semi-stable reduction

Example 4.2.2. To get an example for a curve with semi-stable reduction andnon-regular components, we look at the Weierstraß-equation η2ζ− ξ(ξ− ζ)(ξ−λζ) defining an elliptic curve X in P2

K for a λ ∈ K with |λ| < 1 and coordinates(ξ, η, ζ). The standard affinoid cover of P2

K with three copies of B2K gives the

nodal curve defined by T 20 T2−T 2

1 (T1−T2) in P2k as the reduction of X. Blowing

up (ξ, λ) in this model and changing coordinates in a suitable manner gives usthe curve of Example 1.11.4. The reduction of X now has two componentsX1, X2 of genus 0, which are regular.

Definition 4.2.3. For a semi-stable curve X, we want to define the dual graphG(X). The vertex set V of G is the set of irreducible components of X. Theedge set E of G is the set of double points of X, where an edge e connects thevertices corresponding to the components which intersect in the correspondingdouble point.

For a semi-stable curve X we will denote the double points corresponding tothe edges εi (as in Proposition 4.1.3) by yi.

Example 4.2.4. The dual graph G(X) of the elliptic curve of Example 1.11.4has two vertices and two edges between these edges. Removing one of theseedges yields a tree. We denote the double point corresponding to this edge asy.

We now apply the results for graphs of Section 4.1 to a p-adic curve X withsemi-stable reduction X.

Proposition 4.2.5. Let X be a smooth projective formal analytic curve withsemi-stable reduction X. Let G(V,E) be the dual graph of X. For any abeliangroup A we have

Hq(X,A) = Hq(X, A) = Hq(G,A) .

Proof. Let e ∈ E be an edge corresponding to a double point x between thecomponents Xi and Xj . Define

Ze := π−1(((

Xi ∪ Xj

)\ Sing X

)∪ x)

.

By our assumption, x is not the only singularity of X, so Z := Ze ; e ∈ Eis an open affinoid covering of X. The part of the graph associated to thereductions Ze are the Ge of the proof of Proposition 4.1.5. This implies thatHq(Z, A) = Hq(Z, A) = Hq(G, A).

To prove that Hq(Z, A) = Hq(X,A) it suffices to show that A is acyclic onthe intersections of elements of Z. This can be done by a case-by-case analysisof the intersections as in [BL84, Prop. 2.2].

Corollary 4.2.6. The isomorphism : H1(G,A) →At gives rise to the iso-morphism ψ : Gt

m,K →H1(X,K×), ψ : Gtm,K →H1(X,R×) and its reduction

ψ : Gtm,k →H1(X, k×). This gives these cohomology groups are canonical struc-

ture as analytic or algebraic groups.

55


4.3. The Jacobian of a semi-stable curve

Let X be a semi-stable curve. As seen in Chapter 3 the Picard group Pic X ofX is defined as the isomorphism classes of line bundles over X. This group iscanonically isomorphic to H1(X,O×

X). In this situation the group CaCl(X) of

Cartier divisors modulo linear equivalence coincides with Pic X.

Proposition 4.3.1. The group Pic X can be represented as Div X/ ∼ whereDiv X are the Weil divisors on X with support in the non-singular locus of Xand ∼ is the usual linear equivalence.

Proof. See [BL84, pg. 274]

Note that our definition of the Jacobian variety in Chapter 3 is only valid fornon-singular curves. We can now generalize this concept.

The Weil divisors with support in the non-singular locus have a well-defineddegree on each irreducible component of X.

Definition 4.3.2. The Jacobian variety Jac(X) of X is the subgroup of Pic(X)composed of the classes of divisors with degree 0 on each component.

Theorem 4.3.3. The natural map H1(X, k×) → H1(X,O×X

) induces a shortstrict exact sequence

1→ Gtm,k

ψ−→ Jac Xρ−→ B :=

∏Jac Xi → 1 (4.2)

Proof. The second morphism ρ is obtained by projecting a divisor pointwiseonto the irreducible components of X, i.e. the pull-back f∗i L of a line bundle Lby the canonical immersion fi : Xi → X. This map is obviously surjective andcompatible modulo principle divisors.

Let now [D] ∈ ker ρ. This means that [D|Xi ] = 1, i.e. D|Xi = (fi) for a

rational function fi on Xi. Since D has support in the non-singular locus, everyfi is defined and non-zero in every double point x of X. Let x1, x2 ∈ Xi be twodouble points on the same component of X with corresponding edges e1, e2 witha common vertex v in the dual graph of X. Setting αe1,e2 = fi(x1)/fi(x2) yieldsan element of H1(X, k×) which corresponds to the natural map H1(X, k×)→H1(X,O×

X).

On the other hand take an element α ∈ H1(X, k×). We know by a well-knowncorollary of the Riemann–Roch Theorem that there exists a function with givenvalues in the finitely many double points of Xi, so we can set the values at thegiven double points as we wish. Therefore every element of H1(X, k×) yieldsappropriate functions as before. Since every component of X is a projectivecurve, the divisor corresponding to these function will have degree 0 on eachcomponent. This proves the exactness of the sequence.

To see that (4.2) is strict exact, we have to look at the geometric structure ofJac X. Without restriction, we can assume that every marked double point xiis blown up to a component isomorphic to P1

k, intersecting the rest of X in thepoints at zero and infinity. A collection (fi) of functions can be renormed at

56


the unmarked double points and represented by (ciZ−1)/(Z−1) for a suitableci ∈ Gm,k on the blown up P1

k with coordinate Z for the marked double pointxi, yielding a rational function on all of X.

This way, we can set

X(∗)′ = Gtm,k ×

n∏i=1

X(gi)i

where X(gi)i is the gith symmetric product of the ith component with gi as

the genus of that component. Thereby we can construct the Jacobian varietywith the usual Weil construction, see for example [Mil86, §7]. Since

∏Jac Xi

is constructed the same way, we see that ψ embeds Gtm,k as a closed subvariety

and that ρ is smooth.

We can now directly describe the group law of the Jacobian by the methodsof Chapter 3. We need to make the Weil construction of the Jacobian moreexplicit. We select a non-singular base point xi for every component Xi of X.

Then Jac Xi has an open subset Ui isomorphic to an open subset of X(gi)i where

gi is the genus of Xi given via the morphism

X(gi)i → Jac Xi ; D 7→ [D − gixi]

We can without restriction assume that Ui has no support in the singular pointsof X. On Ui, the group law is given via D1 + D2 = div hi + D1 + D2 − gxiwhere hi is the global section of L(D1 + D2 − gxi), unique up to a constant.

Let X(∗) =∏X

(gi)i and let U be the product of the subsets Ui. On U we can

define a section of ρ by sending D ∈ X(∗) to [D −∑gixi] on Jac X. This map

defines the open subset Gtm,k × U of Jac X with the group law

(a1, D1) + (a2, D2) = (a1a2f(D1, D2), D1 +D2) .

The cocylce f is defined by selecting the hi so that they agree on the unmarkeddouble points and then taking the quotient of the two values at the double pointcorresponding to εi as the ith component as described in the proof of Theorem4.3.3.

Proposition 4.3.4. Let X ′ be a regular curve of genus g and x1, x2 two pointson X ′. Denote by X the semi-stable curve obtained by identifying x1 and x2 onX ′. If x1 = x2 this gives a rational nodal curve glued with X ′ in x1. The linebundle of Proposition 3.5.4 corresponding to the extension (4.2) is given by thepoint [x2− x1] of Jac X ′ regarded as a line bundle on Jac X ′ via the autodualityJac X ′ → Pic0(Jac X ′) given by any base point x0.

Proof. Proposition 3.5.4 states that (4.2) means that E := Jac X is a Gm,k-torsor associated to a line bundle over B := Jac X ′.

Recall that this line bundle is given by a rational section s : U → E of ρ in

0→ Gm,kψ−→ E

ρ−→ B → 0

57


together with its translations sc : c+ U → E as sc(x) = s(x− c) + s(c) for anyc ∈ U . These give isomorphisms

ϕc : Gm,k × (c+ U)→ ρ−1(c+ U)

by ϕc(a, u) = ψ(a) + sc(u).

A rational section t of ρ gives rise to a rational function tc = p1 ϕ−1c t for

tc : c + U → P1. By the definition of ϕc we get tc = p1 (t − sc). Since polesand zeroes of tc behave well when changing c this gives the Cartier divisor onC that belongs to the extension.

So to calculate the divisor associated to JacX as a line bundle we only needto determine the divisor of s.

With [Mil86, Lemma 6.9./ Remark 6.10.] we know that

ι∗ : Picτ (Jac X ′)→ Pic0 X ′

is an isomorphism, where

ι : X ′ → Jac X ′ ; x 7→ [x− x0]

is the canonical embedding with respect to x0.

So when we fix a point x0 on X ′ any divisor class on Jac X ′ is defined byan element of Jac X ′. This leaves us with calculating the divisor of s ι asdescribed above.

Let U ⊂ (X ′ \ x1, x2)(g) be the set of effective divisors with degree g suchthat D− gx0 is non special for all D ∈ U . Since divisors of the form x− x0 arealways non special we see that ι|X′\x1,x2 is restricted to U .

We get a rational section s : U → Jac X of (4.2) by sending [D − gx0] ∈Jac X ′ = B to [D− gx0] ∈ Jac X = E. Choose any Dc ∈ U such that Dc 6= gx0

and that Di ∈ U where Di is defined by

[xi − x0] + [Dc − gx0] = [Di − gx0] ,

for i = 1, 2.

If D is any effective divisor in U ∩ Dc + U we can define D′ as the uniqueeffective divisor that fulfills

[D − gx0]Jac X′ − [Dc − gx0]Jac X′ = [D′ − gx0]Jac X′ (4.3)

and we get

sc([D − gx0]Jac X′) = p1 ϕ−1c (s([D − gx0]Jac X′ ]))

= p1

(s([D − gx0]Jac X′

)− sc

([D − gx0]Jac X′

))= p1

(s([D − gx0]Jac X′

)−s([D − gx0]Jac X′ − [Dc − gx0]Jac X′

)−s([Dc − gx0]Jac X′

)).

58


We can now use 4.3 to simplify the second summand to get

= p1

(s([D − gx0]Jac X′)− s([D

′ − gx0]Jac X′)

−s([Dc − gx0]Jac X′))

= p1([D − gx0]Jac X − [D′ − gx0]Jac X − [Dc − gx0]Jac X)

= hD(x1)

where hD ∈M′X is a rational function with the divisor

div hD = D′ +Dc −D − gx0

and hD(x2) = 1 by the definition of s and ψ.

We can interpret the hD as a rational function h on JacX ′ × X ′. To findthe divisor associated to the extension, we need to find the zeroes and poles ofh(·, x1). By applying ι we can interpret h as a rational function on X ′×X ′ andreduce the problem to finding the zeroes and poles of the first argument.

We can choose Dc in such a way that the support of Dc, D1 and D2 doesnot contain x1 and x2. Using the definition of h on X ′ × X ′ we see that thediagonal is a simple pole. This is the only component of the divisor that meetsthe point (x1, x1) by our assumption on Dc.

Therefore h(ι(x), x1) has a single pole at x1 and by the same argument asingle zero at x2.

This means that JacX is a total fiber space of the divisor class [x2− x1] overJac X ′ for any base point x0 ∈ X ′.

If x1 = x2, we see that the rational functions on X are the same as on X ′,thus the extension corresponds to the trivial line bundle.

Proposition 4.3.5. Let X be a semi-stable curve. Denote by xi,γ,1 and xi,γ,2the two double points of the component Xi which lie on the cycle γ with thecorresponding orientation. The line bundles given by the extension (4.2) aregiven by Lγ =

⊗[xi,γ,2 − xi,γ,1].

Proof. We can use the same proof as in 4.3.4 by replacing X ′ with the disjointunion of the irreducible components Xi of X. We fix a base point xi for everyirreducible component and let s be the section induced by these base points, i.e.s(⊗

[Di − gixi]Jac X′) =⊗

[Di − gixi]Jac X for suitable effective divisors Di on

Xi. We can choose a divisor Di,c with support solely in Xi to translate s andthereby calculate the divisor of s on the component Xi by the same method asin 4.3.4 which gives Lγ |Xi = [xi,γ,2 − xi,γ,1]. This proves the assumption.

Proposition 4.3.6. The sequence (4.2) of Theorem 4.3.3 will split if and onlyif the vertices belonging to the simple curves γi are rational curves.

Proof. By the virtue of Proposition 4.3.5, we only need to determine when thebundle [x2 − x1] on every component Xi is trivial. If x1 6= x2 the triviality ofthe bundle gives an isomorphism of Xi to P1, so Xi is rational.

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4.4. The Jacobian of a curve with semi-stable reduction

To apply this result to the original curve X, we lift the exact sequence ofTheorem 4.3.3.

We want to describe the set of Weil divisors analogous to the Weil divisorsused in Definition 4.3.2.

Definition 4.4.1. We say that a Weil Divisor D on X has support in thenon-singular locus of the reduction if we have suppD ⊂ π−1(X \ Sing X).

We say that D has degree 0 in the reduction if D has degree 0 on everyirreducible component of X.

The Divisors with support in the non-singular locus of the reduction anddegree 0 in the reduction are denoted by DivX

With [BL84, Thm. 5.1] we know that DivX/ ∼ yields an open analyticsubgroup J of JacX with the canonical reduction Jac X.

Theorem 4.4.2. There is a strict exact sequence

1→ Gtm,K

ψ−→ Jρ−→ B → 1 (4.4)

which reduces to (4.2) of Theorem 4.3.3.

Proof. See [BL84, Theorem 6.6].

Note. If the sequence (4.4) splits, then the exact sequence (4.2) of the reductionwill also split, since the splitting morphism can be reduced. On the other handknowing that the sequence of the reduction splits is not sufficient to deducewhether or not sequence (4.4) splits.

We want to extend this exact sequence to the whole Gtm,K . This can be done

via the theory of group objects as we have seen in the previous chapter.

Theorem 4.4.3. There is a exact sequence

1→ Gtm,K

ψJ−−→ J → B → 1 (4.5)

withJ :=

(Gtm,K × J

)/((g, j) ∈ Gt

m,K × J ; ψ(g) = j−1).

Proof. There is a unique group homomorphism ϕ : Gm,K → Gm,K , so J is justthe push forward ϕ∗J .

The projection on the first factor of the direct product gives a surjectivemorphism

ϕ : J → |K×|t → Rt ; (g, j) 7→ |g| 7→ − log|g| . (4.6)

In the rest of this section we will discuss how we can use the analytic groupJ to describe the Jacobian JacX.

Definition 4.4.4. Let OX be the sheaf of functions with supremum norm 1 onevery component. Let O×X be the subsheaf of functions without zeros.

60


Definition 4.4.5. With the natural morphisms, we can regard H1(X,K×) andH1(X, O×X) as part of H1(X,O×X). We call the bundles coming from H1(X,K×)

toric and the bundles coming from H1(X, O×X) formal. Note that the bundlesin H1(X,R×) are both toric and formal.

Lemma 4.4.6. Every element f ∈ H1(X,O×X) can be written as a product

f = α·g of a toric bundle α ∈ H1(X,K×) with a formal bundle g ∈ H1(X, O×X).

Proof. Let f = (fi,j) ∈ H1(X,O×X) be represented by a cocycle on an formalopen covering U = (Ui)i∈I which is a refinement of the covering Z. With theappropriate blow-up and further refinement we can achieve that Ui ∩ Uj ⊂π−1

(Xk \ Sing X

)for every i 6= j. Blowing up double points of the reduc-

tion only adds curves of genus 0, on which every divisor is trivial. Hence thecohomology group H1(X, O×X) remains unchanged.

We know that the supremum norm is multiplicative on intersections Ui ∩Uj ⊂ Xk := π−1

(Xk

)and thus that the absolute value of fi,j is well defined.

We can find a cocycle α = (αi,j) ∈ C1(U,K×) with |αi,j | = |fi,j |. Settinggi,j = fi,j/αi,j gives us a cocycle g with absolute value 1 on each component,i.e. g ∈ C1(U, O×X).

We now calculate these factorizations explicitly for line bundles correspondingto divisors of the form [x1 − x2], x1, x2 ∈ X. For a line bundle L denote by DLthe corresponding divisor and by fL ∈ H1(X,O×X) the representing element ofthe cohomology group.

Lemma 4.4.7. An analytic Weil divisor with support in the non-singular locusof the reduction can be solved locally and thus be written as a Cartier divisor.

Proof. Let x ∈ X be a point with the regular point x as reduction. We canfind a local parameter of x, i.e. a function f ∈ OX(U) which has only one zero

on U in x and ordx f = 1. A lift f of f can only have a single zero in X+(x)and has |f(y)| < 1 for all y ∈ X+(x) according to Corollary 2.2.7. Thereforef − f(x) ∈ OX(U) solves the divisor [x] locally on U = π−1(U).

Proposition 4.4.8. Let x1, x2 ∈ Xi := π−1(Xi \ Sing X) be two points onthe same component of X. Then the line bundle L corresponding to DL =[x1 − x2], is represented by an element fL = (fi,j) ∈ H1(X,O×X) that is already

in H1(X, O×X).

Proof. The set U1 = X \ π−1(x1, x2) is formal open on X and since L istrivial on U1 it can be represented by the constant function m1 = 1. Since Lhas no support outside of Xi, we can choose a formal open covering U2, . . . , Urof Xi and meromorphic functions mi on the Ui which represent DL by Lemma4.4.7. Without restriction, we can set the absolute value of mi to 1, since thesupremum norm is multiplicative on Xi. Therefore the line bundle L will berepresented by (mi) and the transformation functions fi,j = mj/mi on Ui ∩ Ujhave absolute value 1. Thus f ∈ H1(X, O×X).

Corollary 4.4.9. The analytic group J is a subgroup of H1(X, O×X).

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Proposition 4.4.10. The map

η : J → JacX ; (g, j) 7→ g · j

is well-defined and a surjective morphism of analytic groups.

Proof. In order to show that the morphism η is well defined, we need to showthat every toric line bundleM has degree zero. But the connected variety Gt

m,K

parameterizes the toric line bundles, so their degree cannot change and thetrivial bundle has degree zero. By Corollary 4.4.9 we know that J is a subgroupof H1(X, O×X) and the canonical map H1(X, O×X)→ H1(X,O×X) coincides withthe natural embedding J → JacX for bundles of degree zero. The equivalencerelation ψ(g) = j−1 for g ∈ Gt

m,K is corresponding to the fact that H1(X,R×)

is a subgroup of both H1(X,K×) and H1(X, O×X).

By Lemma 4.4.6 we can write any line bundle L of degree zero as a productof a toric bundle M and a formal bundle N . As we already noted, the toricbundle M has degree zero which implies that N has degree zero as well and isin J . Hence η is surjective.

Note. The morphism η is not injective as there are line bundles that are bothformal and toric and yet not induced by R× as we will see later in this chapter.

We can assume without restriction that the dual graph contains no loops, aswe have discussed in the introduction of this section. This leaves only the nextproposition as second case to evaluate.

Proposition 4.4.11. Let e ∈ E be an edge with corresponding double pointx, lying on components Xi and Xj, where Xi corresponds to the source of eand Xj corresponds to the target of e. Let x1 ∈ π−1(Xi \ Sing X) and x2 ∈π−1(Xj \ Sing X). We denote by q the height of the annulus of the formal fiberof x.

Then the line bundle L with DL = [x1 − x2] can be factored into α · g withα = α(e, q) ∈ H1(X,K×), the weighted cocycle of q in e as in Definition 4.1.6and g ∈ H1(X, O×X).

Proof. Let U1 be a formal neighborhood of π−1(x). We have seen in Chapter 2that the formal fiber of x is an annulus with parameter ζ such that |ζ(x)| = 1for x ∈ Xi and |ζ(x)| = q for x ∈ Xj . By proposition 4.4.8 we can adjust x1

and x2 on their respective component so that both x1 and x2 are in U1. ThenL can be represented on the interior of U1 by the meromorphic function

m1 =ζ − x1

ζ − x2,

which expands uniquely to U1. On U2 := X\π−1(x, x1, x2) we choose m2 = 1.Then U = U1, U2 is a formal open covering of X and the mi represent DL.

m1 has absolute value 1 on Xi and absolute value q−1 on Xj . So m2/m1 hasabsolute value 1 on Xi and q on Xj . Refining U as in Lemma 4.4.6 thereforegives us the toric line bundle α(e, q) in the factorization of f .

62


Recall that we have defined

ϕ : J → Rt

by

ϕ(a, g) = − log|a| .

To further describe η we need to look at its kernel. As it turns out this kernelis a lattice in J i.e. a discrete subgroup. Such a group is mapped to a latticein Rt under ϕ, retaining most of the important information. The next theoremuses the previous two propositions to give a basis for that lattice. In the proofwe will also construct the kernel of η without applying ϕ but this constructionis rather abstract, since the formal factor is left implicit.

Theorem 4.4.12. The group ϕ(ker(η)), the projection of first factor of thekernel of the morphism η of Corollary 4.4.10 is a lattice of rank t in Rt. If thesimple cycles of Proposition 4.1.4 are denoted by γi, then this lattice is generatedby vi = (vij)

tj=1 with

vij =∑

e∈γi∩γj

−d(e) · log|q(e)|

where q(e) is the height of the formal fiber of the double point correspondingto e and d(e) = 1 if e has the same orientation in γi as in γj and d(e) = −1otherwise.

Proof. We factorize the trivial line bundle in a non trivial way for each cycleγ = (v1, . . . , vm), vm+1 = v1 in G(X). Select a point xj on each componentXj := π−1(Xj \Sing X) corresponding to vj . We know that

⊗mk=1[xk−xk+1] ∼=

O. So by using Proposition 4.4.11 we get a toric factor M = (αij). Thecorresponding element a = (ai) ∈ Gt

m,K of α is easily computed with Lemma4.1.7 to be

ai =∏

e∈γi∩γq(e)d(e)

where q(e) is the height of the formal fiber of the double point correspondingto e and d(e) is the multiplicity of e in γ counted positive if the direction of ein γ and γi coincide, negative if not. Note that the cycles γi are simple, so e isonly found once in γi.

Since every cycle can be written as a formal sum of the γi, each factorizationobtained by this procedure will have an image in the given lattice.

Furthermore we obviously get ϕ(J) = 0 and η−1(J) ∩ J = J , since ϕ wasdefined this way and η is an isomorphism when restricted to J . So ϕ(ker η) isa lattice and we can concentrate on the first factor of every factorization of thetrivial line bundle.

Suppose there is another factorization O =M⊗N of the trivial line bundleinto a toric line bundle M and a formal line bundle N . We represent thesefactors by the cocycles (αij) and (gij) on a formal open covering U as in 4.4.6,i.e. U refines Z and Ui ∩ Uj ⊂ Xk := π−1(Xk \ Sing X). This means that foreach Ui ∈ U, we have an fi ∈ O×X(Ui) with αijgij = fj/fi on Ui ∩ Uj .

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We have two cases to consider. There are Ui for which Ui ⊂ Xk holds forsome k. In this case we can set |fi| = 1, changing α only by a coboundary.The other case we have to consider is Ui ⊂ Ze with non empty intersectionwith both components of Ze. For each e ∈ E there is only one such Ui sincewe have chosen the refinement of U accordingly. Denote by Xk the componentcorresponding to the source of e and by Xj the component corresponding tothe target of e. We can adjust fi such that |fi|Xk | = 1. Then fi, the reductionof fi on Xk, is well defined. Set ne := ordx fi the order of fi in x the doublepoint corresponding to e. We have |fi|Uj | = qne .

Select an arbitrary point of non singular reduction xk on Xk and xj on Xj .By Proposition 4.4.11 the line bundle Le corresponding to the divisor classDe = [−nexk + nexj ] will yield the same toric factor on Ui. This means that

L =⊗e∈ELe

can be factored to L =M⊗N ′.We have O,N ,N ′ ∈ J by definition, hence M,L ∈ J . The divisor D =∑e∈E De has support in the non-singular locus of the reduction. We can modify

D with the divisor of a rational function such that degD|Xk = 0 holds for eachcomponent Xk. Modifying D by the divisor of a rational function does notchange the cocycles, since a rational function corresponds to the unit cocyclein both factors. This means that the chain

∑e∈E −nee has zero boundary on

G(X) and is a cycle. Therefore ϕ(M) is already an element of the lattice definedabove.

Proposition 4.4.13. In the case where the sequence (4.4) splits, the lattice ofTheorem 4.4.12 lies completely in the Gt

m,K factor of J i.e. we get

JacX =(Gtm,K/M

)×B

with B being the analytic group variety with good reduction of sequence (4.4)

Proof. We have the following diagram of exact sequences.

1

!!

1

1 // Gt

m,K// J

!!

// B //

σ

1

1 //M // Jη // JacX // 1

Now the injective splitting morphism σ of (4.4) implies an arrow to JacX whichis necessarily injective. Furthermore, in the splitting case we have J = Gt

m,K×Band therefore J = Gt

m,K ×B by its definition. Thus the kernel of η is restricted

to Gtm,K × 0, which gives us a way to see M as a lattice purely in Gt

m,K .

We want to calculate the abelian variety B in a specific example. For this weneed the following lemma.

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Lemma 4.4.14. Let X and Y be two projective smooth curves and ρ : X → Ybe an algebraic surjective map of degree n coprime to charK. Then ρ∗ ρ∗ =n · idJacY , where ρ∗ is the induced map and ρ∗ := (ρ∗)∗ its dual.

Proof. The situation is as follows

X

ρ

JacX∼ // (JacX)′

ρ∗

Y JacY //

ρ∗

OO

(JacY )′ .

Let us look at a more explicit description of ρ∗ and ρ∗. If D =∑

y∈Y nyy

is a divisor, then ρ∗D =∑

y∈Y nyρ−1(y). This definition coincides with the

one of Chapter 3 as one can easily check by writing D as a Cartier divisors.Similarly for Divisors D =

∑x∈X nxx with nx1 = nx2 for all x1, x2 ∈ X with

ρ(x1) = ρ(x2) the map ρ∗(D) is given by ρ∗(D) =∑

x∈X nxρ(x). On cancheck this once again by calculating (ρ∗)∗ using cocylces and the fact thataccording to Torelli’s theorem the map ρ∗ maps the theta structure of JacYonto that of JacX. Clearly ρ∗ ρ∗ = n · id. Another way to put this isthat ρ∗ρ

∗L = N(ρ∗(L)) = L⊗n where N is the norm induced by ρ on thecorresponding function fields.

Proposition 4.4.15. Let X be an analytic curve of genus g with semi-stablereduction X such that the dual graph of X has cyclomatic number 1. Let fur-thermore ρ : X → Y be an algebraic surjective map of degree n such that Yhas a reduction whose dual graph is a tree and genus g − 1. Then the analyticvariety B of (4.4) is isogenous to JacY and J is the total fiber space of ann-torsion point of Pic0(B).

Proof. The dual of the map ρ∗ is ρ∗ : JacX → JacY and has a kernel ofdimension one. The reduced subscheme of this kernel is an elliptic curve E withmultiplicative reduction. We get E = Gm,K and a morphism ψ : Gm,K → Jfrom the embedding. We know there is only one way up to automorphism toembed Gm,K in J so ψ is the ψ of Theorem 4.4.2.

Let ρ : J → B be the cokernel of ψ. Since ρ∗ ψ = 0 we get a homomorphismγ : B → JacY . Since JacY and B have the same dimension g − 1 and sinceρ∗ = γ ρ we see that γ is an isogeny.

We construct σ : B → J as σ = ρ∗ γ. With Lemma 4.4.14 we calculate

γ ρ σ = γ ρ ρ∗ γ= ρ∗ ρ∗ γ= n · idJacY γ .

so ρ σ = n · idB. This means that for a rational section s : B 99K J of ρ we getns = σ up to coboundries. Therefore J is the total fiber space of an n-torsionpoint of Pic0(B).

Note. It should be noted that even if B = JacY is the Jacobian of some curve,the existence of the map ρ : J → B does not imply a map ρ : X → Y , since the

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curve X cannot be embedded in the variety J . In fact it is easy to construct anexample where X has the same reduction properties as in Proposition 4.4.15which does not come from an n-torsion point. Take any line bundle with goodreduction over any elliptic curveB. Dividing by a lattice gives a two dimensionalabelian variety which is the Jacobian of an genus two curve X by Torelli’sTheorem. Since there are line bundles which are not n-torsion points there willbe no elliptic curve on which X can map onto.

4.5. Examples

Example 4.5.1. We will discuss Theorem 4.4.12 in the example of an ellipticcurve X. If X has good reduction the dual graph G(X) contains only one vertexand has cyclomatic number t = 0. We therefore get J = JacX = X as usual.In the case of bad reduction consider the model of Example 1.11.4. This meansthat X has two components of genus 0 meeting in two double points with |√q|as height of the formal fiber. We have exactly one simple cycle in G(X), whichmeans that our lattice is generated by the logarithm of |q| = |√q1 · √q1|. Since

J = Gm,K the sequence (4.4) splits and thus we know that J = Gm,K and thatthere is a point q ∈ Gm,K such that all trivial toric bundles are represented byqZ. And indeed Gm,K/q

Z is equal to JacX = X.

Example 4.5.2. Let us calculate the lattice for curves of genus 2. In case ofgood reduction we have JacX = J as before. For bad reduction, we have threecases to consider.

First we can have t = 0 as cyclomatic number. In this case the reduction ofX consists of two elliptic curves meeting at one ordinary double point. Sincet = 0 the toric part is trivial and JacX = J = B. Describing B and its relationto the elliptic curves of the reduction in this case is rather difficult, some casesare described in [FK91].

Second if the cyclomatic number t equals 1, then X consists of an elliptic curveintersecting a rational curve in two double points of height q1 and q2 respectively.There is only one simple cycle and we see that the lattice is generated bylog|q1q2|. In the splitting case we furthermore get JacX = Gm,K/(q1q2)Z × Ewith an elliptic curve E with good reduction.

The lattice is the most interesting in the total degenerate case of t = 2 whereX consists entirely of rational curves. Since B = 0 we know that the lattice liescompletely in G2

m,K . There are two possibilities for G(X) save for blow-ups.

In the first case G(X) will look like

•e1((

hhe2

•e3((

hhe4

• ,

with corresponding heights q1, q2, q3 and q4. We select γ1 = e1 + e2 and γ2 =e3 +e4. These cycles have no intersection, so the lattice is generated by (q1q2, 1)and (1, q3q4).

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4.5. Examples

In the second case we get the graph G(X)

•

e1

%%ee

e3

oo e2 • ,

with the height of the double points being q1, q2 and q3. We select the simplecycles γ1 = e1 + e2 and γ2 = e1 + e3. γ1 and γ2 intersect at e1 and e1 hasthe same direction in both cycles. This means that the lattice is generated by(q1q2, q1) and (q1, q1q3).

Example 4.5.3. We want to describe B in detail in the case of a special genustwo curve with cyclomatic number 1.

For this let λ, a, b ∈ K with |λ| < 1 and |a| = |b| = 1. We want to describethe Jacobian of the genus two curve X given by

U1 := SpK〈ξ, η〉/(η2 − (ξ2 − λ2)(ξ2 − a2)(ξ2 − b2))

U2 := SpK〈σ, τ〉/(τ2 − (1− λ2σ2)(1− a2σ2)(1− b2σ2))

glued via σ = 1/ξ and τ = η/σ3.The reduction induced by Ui is given by

U1 := Spec k[X,Y ]/(Y 2 −X2(X2 − a2)(X2 − b2))

U2 := Spec k[S, T ]/(T 2 − (1− a2S2)(1− b2S2))

with S = 1/X and T = Y/S3. This curve has a self intersection at X = 0, Y =0. To get rid of this self intersection we blow U1 up by subdividing it in parts|ξ| ≤ λ and |ξ| ≥ λ. We get

V1 := SpK〈χ, ϑ〉/(ϑ2 − (χ2 − 1)(λ2χ2 − a2)(λ2χ2 − b2))

V2 := SpK〈ξ, η, ζ〉/(ζξ − λ, η2 − (1− ζ2)(ξ2 − a2)(ξ2 − b2))

V3 := SpK〈σ, τ〉/(τ2 − (1− λ2σ2)(1− a2σ2)(1− b2σ2))

where χ = λ−1ξ and η is modified accordingly.This gives the reduction X

V1 := Spec k[V,W ]/(W 2 − (V 2 − 1))

V2 := Spec k[X,Y, Z]/(XZ, Y 2 − (1− Z2)(X2 − a2)(X2 − b2))

V3 := Spec k[S, T ]/(T 2 − (1− a2S2)(1− b2S2))

which is the elliptic curve given by the ramification points ±a and ±b gluedwith a rational curve intersecting at X = 0, Y = ±ab. Any divisor with supportin the non singular locus of the reduction is a divisor with support in V1 ∪ V2.Since J is part of the Jacobian which is birational to X(2) we can deduce that

ι : V2(λξ−1, λ−1ξ)× V2(ξ−1)−→ J

((ξp, ηp), (ξq, ηq)) 7−→ [(ξp, ηp) + (ξq, ηq)− (λ, 0)− (a, 0)]

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is a formal birational map describing J .There are two elliptic curves E1 and E2 gained by

U1 := SpK〈ξ1, η1〉/(η21 − ξ1(ξ1 + λ2)(ξ1 + a2)(ξ1 + b2))

U2 := SpK〈σ1, τ1〉/(τ21 − (1 + σ1λ

2)(1 + σ1a2)(1 + σ1b

2))

and

V1 := SpK〈ξ2, η2〉/(η22 − (ξ2 − λ2)(ξ2 − a2)(ξ2 − b2))

V2 := SpK〈σ2, τ2〉/(τ22 − σ2(1− σ2λ

2)(1− σ2a2)(1− σ2b

2))

with σi = 1/ξi and τi = ηi/ξ2i as gluing relations. We also get maps of degree 2

ψ : X → E1 ρ : X → E2

ψ(ξ, η) = (−ξ2, η · ξ) ρ(ξ, η) = (ξ2, η)

These maps induce pullbacks on Pic which give the maps

ψ∗ : E1 → JacX

[(−ξ2p , ηp)− (−a2, 0)] 7→

[(ξp, ηp/ξp) + (−ξp,−ηp/ξp)− (a, 0)− (−a, 0)]

ρ∗ : E2 → JacX

[(ξ2p , ηp)− (a2, 0)] 7→

[(ξp, ηp) + (−ξp, ηp)− (a, 0)− (−a, 0)]

ψ∗ : JacX → E1

[(ξp, ηp) + (ξq, ηq)− (a, 0)− (−a, 0)] 7→[(−ξ2

p , ξpηp) + (−ξ2q , ξqηq)− 2(−a2, 0)]

ρ∗ : JacX → E2

[(ξp, ηp) + (ξq, ηq)− (a, 0)− (−a, 0)] 7→[(ξ2

p , ηp) + (ξ2q , ηq)− 2(a2, 0)] ,

where ψ∗ and ρ∗ are the dual maps of ψ∗ and ρ∗ with (ξp, ηp), (ξq, ηq) as thepreimage of ψ∗(x) or ρ∗(x), extended to JacX uniquely.

As noted in Lemma 4.4.14 we have

ψ∗ψ∗([(−ξ2

p , ηp)− (−a2, 0)]) = ψ∗([(ξp, ηp/ξp) + (−ξp,−ηp/ξp)− (a, 0)− (−a, 0)])

= [2(−ξ2p , ηp)− 2(−a2, 0)]

ρ∗ρ∗([(ξ2

p , ηp)− (a2, 0)]) = ρ∗([(ξp, ηp) + (−ξp, ηp)− (a, 0)− (−a, 0)])

= [2(ξ2p , ηp)− 2(a2, 0)] .

One checks that

ρ∗ψ∗([(−ξ2

p , ηp)− (−a2, 0]) = [(ξ2p , ηp/ξp) + (ξ2

p ,−ηp/ξp)− 2(a2, 0)] = 0 .

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4.5. Examples

Furthermore ψ∗ is injective and ρ∗ is surjective so we get the exact but notstrict exact sequence

0 // E1ψ∗ // JacX

ρ∗ // E2// 0 . (4.7)

To see that (4.7) is not strict exact, look at the fibers of ρ∗. The points [(ξp, ηp)+(ξq, ηq)− (a, 0)− (−a, 0)] that map to zero are given by ξ2

p − ξ2q and ηp + ηq on

X2. This is a singular variety in JacX so ρ∗ is not smooth according to [Har77,III. Lemma 10.5]. The morphism ψ∗ however is indeed a closed immersion.

The curve E1 has multiplicative reduction. Blowing up at |ξ| = |λ2| againresolves the self intersection and leaves us with E1 as the points of E1 witha ξ coordinate with absolute value 1 by mapping (−ξ2

p , ηp) to the line bundle[(−ξ2

p , ηp) − (−a2, 0)]. The map ψ∗ maps this line bundle to the line bundle[(ξp, ηp/ξp) + (−ξp,−ηp/ξp)− (a, 0)− (−a, 0)] which lies in J .

We know by [BGR84, 9.7.] that there is an element q ∈ Gm,K with |q| < 1and an isomorphism ϕ : Gm,K/q

Z → E1 which restricts to ϕ : Gm,K → E1. Thenψ∗ ϕ : Gm,K → JacX has its image in J and is therefore a closed immersionψ : Gm,K → J . Since there is up to automorphism of Gm,K only one injectivemap from Gm,K to J we can without restriction assume that ψ is the ψ ofTheorem 4.4.2.

Let ρ : J → B be the cokernel of this morphism. Since ρ∗ ψ∗ = 0 and ψ is arestriction of ψ∗ we get ρ∗ ψ = 0 and therefore a morphism γ : B → E2. BothB and E2 have dimension 1 and since ρ∗ = γ ρ we see that γ is surjective andis therefore an isogeny.

We can directly calculate that B is Spec k[X,Y ]/(Y 2 − (X2 − a2)(X2 − b2))which is isogenous of order two to E2.

Furthermore we know that ρ∗ ρ∗ = 2 · id and that we get the diagram

Gm,K

ψ

E2ρ∗ //

""

J //

ρ

JacX

B

One calculates that ψ(Gm,K) ∩ ρ∗(E2) consists of 0 and the two torsion point[(b, 0)+(−b, 0)−(a, 0)−(−a, 0)], So γ is an isogeny of order two andB is obtainedby completing SpK〈ξ, η〉/(η2 − (ξ2 − (a2 − λ2)(b2 − λ2)) in the canonical way.

Let’s set σ = ρ∗ γ : B → J . Since

Gm,Kψ // J

ρ //

ρ∗

66Bγ // E2

ρ∗ // Jρ //

ρ∗

66Bγ // E2

we know that ρ σ = ρ ρ∗ γ : B → B is an isogeny. And since ρ∗ ρ∗ = 2 · id

69


we get

γ ρ σ = γ ρ ρ∗ γ= ρ∗ ρ∗ γ= 2 · idE2 γ .

thus ρ σ = 2 · idB.Let s : B 99K J be a rational section of ρ. Then 2s−σ : U → J is in the image

of ψ by the exactness of (4.4) and therefore 2s and σ only differ by a cocycle.But σ is a group homomorphism and therefore defining a trivial bundle. Thismeans that J is the total space of a two torsion point of Pic0(B).

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A. Bibliography

[BGR84] S. Bosch, U. Guntzer, and R. Remmert. Non-Archimedean analy-sis, volume 261 of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences]. Springer-Verlag,Berlin, 1984. A systematic approach to rigid analytic geometry.

[BL84] Siegfried Bosch and Werner Lutkebohmert. Stable reduction anduniformization of abelian varieties. II. Invent. Math., 78(2):257–297,1984.

[BL85] Siegfried Bosch and Werner Lutkebohmert. Stable reduction anduniformization of abelian varieties. I. Math. Ann., 270(3):349–379,1985.

[BLR90] Siegfried Bosch, Werner Lutkebohmert, and Michel Raynaud. Neronmodels, volume 21 of Ergebnisse der Mathematik und ihrer Grenzge-biete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990.

[Bos05] Siegfried Bosch. Lectures on formal and rigid geometry. In SFB 478 –Geometrische Strukturen in der Mathematik, volume 378. Universityof Munster, 2005.

[FK91] Gerhard Frey and Ernst Kani. Curves of genus 2 covering ellipticcurves and an arithmetical application. In Arithmetic algebraic ge-ometry (Texel, 1989), volume 89 of Progr. Math., pages 153–176.Birkhauser Boston, Boston, MA, 1991.

[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York,1977. Graduate Texts in Mathematics, No. 52.

[LT58] Serge Lang and John Tate. Principal homogeneous spaces overabelian varieties. Amer. J. Math., 80:659–684, 1958.

[Mil86] J. S. Milne. Jacobian varieties. In Arithmetic geometry (Storrs,Conn., 1984), pages 167–212. Springer, New York, 1986.

[Mum08] David Mumford. Abelian varieties, volume 5 of Tata Institute of Fun-damental Research Studies in Mathematics. Published for the TataInstitute of Fundamental Research, Bombay, 2008. With appendicesby C. P. Ramanujam and Yuri Manin, Corrected reprint of the second(1974) edition.

[Ray70] Michel Raynaud. Anneaux locaux henseliens. Lecture Notes in Math-ematics, Vol. 169. Springer-Verlag, Berlin, 1970.

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A. Bibliography

[Ser88] Jean-Pierre Serre. Algebraic groups and class fields, volume 117 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1988.Translated from the French.

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Ehrenwortliche Erklarung

Hiermit erklare ich, dass ich diese Arbeit selbststandig angefertigt habe undkeine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie diewortlich oder inhaltlich ubernommenen Stellen als solche kenntlich gemacht unddie Satzung der Universitat Ulm zur Sicherung guter wissenschaftlicher Praxisin der jeweils gultigen Fassung beachtet habe.

Ulm, der . Juni 2013

Sophie Schmieg

73

Curriculum vitae

Personliche Daten

Name: Sophie Schmieg

Geburtsdatum: 4. September 1985

Geburtsort: Ebersberg

Staatsangehorigkeit: deutsch

Ausbildung

seit April 2009 Promotion in Mathematik an der Universitat Ulm imInstitut fur Reine Mathematik

Marz 2009 Diplom in Mathematik mit Nebenfach Informatik(Thema: Zur Theorie der kompakten RiemannschenFlachen und ihrer Jacobi-Varietaten)

Oktober 2006 Vordiplom in Mathematik mit Nebenfach Informatik

seit Oktober 2004 Studium der Mathematik mit Nebenfach Informatikan der Universitat Ulm

Juni 2004 Abitur am Gymnasium Grafing

Sophie Schmieg

75

Rigid analytic curves and their Jacobians

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