Oberseminar Algebraische Geometrie (WS 2011/12)

Prof. Dr. Werner Bley - Prof. Dr. Ulrich Derenthal - Prof. Dr. Andreas Rosenschon

Winter 2011/12, Wednesdays, 16:30, Raum B 251 (Mathematisches Institut, Theresienstr. 39, 80333 München)

Previous semester: Summer 2011 - next semester: Summer 2012

Abstracts

Peter Bruin (Universität Zürich): Ranks of elliptic curves with prescribed torsion over number fields (9.11.11)

Let d be a positive integer, and let T_d be the set of isomorphism classes of groups occurring as the torsion subgroup of E(K), where K is a number field of degree d and E is an elliptic curve over K. It is known that T_d is finite. The sets T₁ and T₂ are known, as well as the subsets of T₃ and T₄ consisting of torsion groups occurring for infinitely many E/K up to isomorphism.

We study the following problem: for d ≤ 4 and T in T_d, what are the possibilities for the rank of E(K) if K is a number field of degree d and E is an elliptic curve over K with torsion group T? In the cases d = 2 and T = Z/13Z or T = Z/18Z, and conjecturally also d = 4 and T = Z/22Z, it turns out that the rank is always even. We explain this by a phenomenon that we call false complex multiplication. This is work in progress with Johan Bosman, Andrej Dujella and Filip Najman.

Jürgen Ritter (Universität Augsburg): Äquivariante Iwasawatheorie (30.11.11)

Nach kurzer Wiederholung der klassischen Iwasawatheorie und ihrer Bedeutung für analytisch-algebraische Zusammenhänge in der Zahlentheorie wird eine nichtabelsche Verallgemeinerung der sogenannten "Hauptvermutung" vorgestellt, die erlaubt, Iwasawatheorie auch in nichtkommutativen Situationen einzusetzen. Der Beweis dieser "Hauptvermutung der äquivarianten Iwasawatheorie" wird grob skizziert werden (und ist gemeinsamer Arbeit mit A. Weiss entnommen).

Jakob Stix (Universität Heidelberg): On the divisibility of the Tate-Shafarevich group of an elliptic curve in the Weil-Chatelet group (7.12.11)

In this talk I will report on progress on the following two questions, the first posed by Cassels in 1961 and the second considered by Bashmakov in 1974. The first question is whether the elements of the Tate-Shafarevich group are infinitely divisible when considered as elements of the Weil-Chatelet group. The second question concerns the intersection of the Tate-Shafarevich group with the maximal divisible subgroup of the Weil-Chatelet group. This is joint work with Mirela Ciperiani.

Arati Khedekar: Cohomology theory for topological groups (11.1.12)

We define an explicit cohomology theory for topological groups based on locally continuous measurable cochains. We study its functorial properties and explore its relationship with other explicit cohomology theories. We apply the theory to the case of connected Lie groups and certain modules and see that extension of a connected Lie group by coefficient modules of certain kind are connected Lie groups.

Michael Dettweiler (Universität Bayreuth): Starre lokale Systeme und Motive (25.1.12)

Wir beleuchten verschiedene Begriffe von starren lokalen Systemen und stellen Methoden zur Konstruktion von assoziierten Motiven vor.

Tamás Szamuely (Alfréd Rényi Institute of Mathematics): Die prim-zu-p Fundamentalgruppe algebraischer Gruppen (1.2.12)

Nach einem klassischen Satz von Schreier ist die Fundamentalgruppe einer topologischen Gruppe kommutativ, und weiterhin hat jede Überlagerung eine Gruppenstruktur. Zusammen mit Michel Brion haben wir die algebraischen Analoga dieser Sätze für prim-zu-p étale Überlagerungen beliebiger algebraischen Gruppen gezeigt.