UNIVERSITÄT AUGSBURG

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UNIVERSITÄT AUGSBURG INSTITUT FÜR MATHEMATIK Universitätsstraße 14 D-86135 Augsburg

Transcript of UNIVERSITÄT AUGSBURG

Page 1: UNIVERSITÄT AUGSBURG

UNIVERSITÄT AUGSBURG

INSTITUT FÜR MATHEMATIK

Universitätsstraße 14 D-86135 Augsburg

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Institut für Mathematik Hausadresse: der Universität Augsburg Universitätsstraße 14 D-86159 Augsburg Geschäftsführender Direktor Telefon (0821) 598-2138 Telefax (0821) 598-2458

e-mail [email protected]___________________________________________________________________________________

Briefadresse: Universität Augsburg, D-86135 Augsburg Augsburg, im Februar 2003

Vorwort zum Jahresbericht 2002 Wie im Vorjahr hatte unser Institut im Jahr 2002 erneut eine deutliche Steigerung der Anfängerzahlen in den Mathematikstudiengängen zu verzeichnen, was der allgemeinen Tendenz in allen Lebensbereichen auf gesteigerte Qualität zu achten, parallel läuft. Das Institut geht davon aus, dass dies eine nachhaltige Entwicklung ist, die in den nächsten Jahren in einen Boom in den Grundlagenforschungen einmünden dürfte. In der Forschung hat unser Institut einen großen Schritt im Ranking der deutschen Forschungslandschaft gemacht. Im Frühjahr wurde gegen harte Konkurrenz aus andren Bereichen ein neuer DFG-Schwerpunkt im Bereich Globale Differentialgeometrie mit Zentrum in Augsburg etabliert. Neben einem bereits bestehenden Schwerpunkt in der komplexen Geometrie ist dies der einzige derart ausgedehnte Forschungsverbund in der reinen Mathematik in Deutscland. Abschließend gilt der Dank des Instituts all denjenigen, die durch ihre Aktivitäten in Lehre und Forschung sowie bei der Einwerbung von Drittmitteln und der Gewinnung von neuen Studenten und wissenschaftlichem Nachwuchs unser Institut für Mathemtik stärken. Ausdrücklich einschließen in diesen Dank möchte ich unsere Sekretariatdienste und technischen Mitarbeiter. Professor Dr. Joachim Lohkamp

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Prof. Dr. Kristina Reiss Lehrstuhl für Didaktik der Mathematik Mathematisch-Naturwissenschaftliche Fakultät Universität Augsburg

Universitätsstr.10, 86159 Augsburg Telefon (0821) 598 – 2494 Sekretariat (0821) 598 – 2492 Fax (0821) 598 – 2278 Email: [email protected]

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Forschungsgebiete am Lehrstuhl fur Algebra und Zahlentheorie

Der Schwerpunkt der am Augsburger Lehrstuhl fur Algebra und Zahlentheorie durchgefuhr-ten Forschungsarbeiten liegt im Beruhrungsfeld der Arithmetik und der Darstellungstheorieendlicher Gruppen, welche in aller Regel als Galoisgruppen von Erweiterungen globaler oderlokaler Zahlkorper erscheinen. Die Veroffentlichungen reihen sich damit in die heute allge-mein im Zentrum des Interesses stehenden zahlentheoretischen Untersuchungen ein und lie-fern Beitrage zur Verifikation und Verfeinerung von grundlegenden Vermutungen, die innerearithmetische Zusammenhange zu beschreiben versuchen.

Die Forschungsarbeit betrifft vornehmlich die weiter unten einzeln vorgestellten Spezialgebie-te. Als Motivation sei ein Blick auf die Fermatsche Gleichung

xp + yp = zp mit ganzen Zahlen x, y, z 6= 0 und Primzahlexponent p ≥ 3

geworfen (deren uber 300 Jahre vermutete Unlosbarkeit erst 1994, von A. Wiles, bewiesenwurde). Sie laßt sich unter Verwendung einer primitiven p-ten Einheitswurzel ζ (etwa ζ =e

2πip ∈ C) in die Produktgleichung

p−1∏i=0

(x + ζiy) = zp

verwandeln, die innerhalb der

ganzen Zahlen oK = p−2∑i=0

aiζi| ai ∈ Z des Zahlkorpers K =

p−2∑i=0

biζi| bi ∈ Q ⊂ C

mit Hilfe von Teilbarkeitsbetrachtungen untersucht werden kann. Nun muß in oK allerdingskeine eindeutige Primzahlproduktdarstellung gelten, womit uns ein erstes Hindernis (mitNamen Idealklassengruppe) in den Weg gelegt ist; des weiteren sind Teilbarkeitsaussagendadurch schwacher als Gleichheiten, daß Einheitsfaktoren, wie etwa 1 + ζ + ζ2 + · · · + ζi

mit 1 ≤ i ≤ p − 2, unberucksichtigt bleiben, woraus ein zweites Hindernis (mit NamenEinheitengruppe) entsteht. Die so aufkommenden Komplikationen werden jedoch durch dasVorhandensein gewisser Symmetrien, namlich Galoissymmetrien, gelindert.

Galoismodulstrukturen

Unter diesen Begriff fallen alle Untersuchungen, die mit der Aufdeckung ganzzahliger Ga-loisstrukturen, wie der des Rings der ganzen Zahlen, der der Einheiten- oder der der Ideal-klassengruppe eines Zahlkorpers K, befaßt sind, sofern K als galoissche Erweiterung einesTeilkorpers k vorliegt. Die beschreibenden Daten werden von analytischen Funktionen, wieetwa Artinschen L-Reihen, vermittelt und zwar meist als spezielle Werte. Dies ist eine uberra-schende Tatsache, die z.Zt. noch nicht voll verstanden wird und deren erste Beobachtung vorca. 35 Jahren an Hand konkreter Beispielrechnungen zu Vermutungen fuhrte, die zunachst als“crazy ideas” abgetan wurden. Das systematische Studium von Analogien zwischen arithmeti-schen und analytischen Eigenschaften im Zusammenhang mit der genannten Problemstellunghat sich aber inzwischen als sehr fruchtbar erwiesen und schone und tiefe Ergebnisse hervor-gebracht. Die wesentlichen algebraischen Ingredienzien kommen dabei aus der ganzzahligenDarstellungstheorie; diejenigen aus der Zahlentheorie schließen die sogenannte Hauptvermu-tung der Iwasawatheorie ein und haben sogar zu Verallgemeinerungen davon gefuhrt. Neue,

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74

mit Blick auf die Galoisstruktur der globalen Einheiten eingefuhrte Invarianten und derenvermutete Eigenschaften fuhren daruber hinaus hin zu den beruhmten Vermutungen uberL-Werte aus der arithmetischen Geometrie.

Komplexe Multiplikation

Elliptische Kurven waren nicht nur beim Beweis der Fermatschen Vermutung ein bedeuten-des Hilfsmittel; inzwischen spielen sie auch in der Kryptographie eine nutzliche Rolle, weilsie eine naturliche Gruppenstruktur besitzen und sich die Koordinaten ihrer Torsionspunktedurch algebraische Gleichungen beschreiben lassen. Allerdings hat bisher die astronomischeGroße der dabei auftretenden Zahlen eine Anwendung verhindert. Wie sich nun in letzterZeit gezeigt hat, lassen sich die Koordinaten der Torsionspunkte durch Konstruktion ge-eigneter Funktionen auf algebraische Gleichungen mit bemerkenswert kleinen Koeffizientenzuruckfuhren. Fur einen Punkt der Ordnung 34 auf der Kurve y2 = 4x3 − 152x + 361 wirddies durch die folgende Gleichung geleistet :

X27 +(−9−

√−19

2

)X26 +

(−11−9

√−19

2

)X25 +

(−113+5

√−19

2

)X24 +

(−197−

√−19

2

)X23

+(

497+77√−19

2

)X22 +

(14− 219

√−19

)X21 +

(−1507−121

√−19

2

)X20 +

(−3853−313

√−19

2

)X19

+(908 + 839

√−19

)X18 +

(−1019−1582

√−19

2

)X17 +

(−10159+5715

√−19

2

)X16 +

(13307− 2428

√−19

)X15

+(−38379+2225

√−19

2

)X14 +

(38379+2225

√−19

2

)X13 +

(−13307− 2428

√−19

)X12 +

(10159+5715

√−19

2

)X11

+(1019− 1582

√−19

)X10 +

(−908 + 839

√−19

)X9 +

(3853−313

√−19

2

)X8 +

(1507−121

√−19

2

)X7

+(−14− 219

√−19

)X6 +

(−497+77

√−19

2

)X5 +

(197−

√−19

2

)X4 +

(113+5

√−19

2

)X3

+(

11−9√−19

2

)X2 +

(9−√−19

2

)X − 1 = 0

Page 79: UNIVERSITÄT AUGSBURG

75

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Rechnerorientierte Statistik und Datenanalyse

Prof. Antony Unwin, Ph.D.

Anschrift

Universität Augsburg Institut für Mathematik

D-86135 Augsburg

Telefon: (+49 821) 598 - 22 18 Telefax: (+49 821) 598 - 22 80

Internet:

[email protected] www1.math.uni-augsburg.de

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