Oberseminar Algebraische Geometrie (SS 2011)

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

Summer 2011, Wednesdays, 16:15, room B 133 (Mathematisches Institut, Theresienstr. 39, 80333 München)

Previous semester: Winter 2010/11 - next semester: Winter 2011/12

Abstracts

Jan Steffen Müller (Universität Hamburg): Algorithmische Aspekte der Vermutung von Birch und Swinnerton-Dyer (15.6.11)

Die Vermutung von Birch und Swinnerton-Dyer liefert eine Verbindung zwischen verschiedenen arithmetischen Größen einer abelschen Varietät und gilt als eines der wichtigsten ungelösten Probleme der arithmetischen Geometrie. Nach der Formulierung der Vermutung wird in diesem Vortrag erläutert, wie man die auftretenden Größen praktisch berechnen und damit die Vermutung numerisch in Beispielen überprüfen kann. Außerdem werden eine p-adische Version der Vermutung für elliptische Kurven und eine mögliche Verallgemeinerung auf modulare abelsche Varietäten vorgestellt.

Marco Seveso (Università di Milano): Families of modular forms, p-adic interpolation and p-adic Gross-Zagier type formulas (22.6.11)

Let F be a Coleman family of modular forms of level N and let F_k be its weight k+2 specialization. Up to a trascendental period, the special values of the complex L-functions L(F_k,χ,s) (resp. L(F_k/K,χ,s)) at s=k/2+1, where χ is a Dirichlet character (resp. a ring class character of a quadratic field K) are known to be algebraic. Under suitable sign conditions, we discuss the problem of p-adic interpolating them into a rigid analytic function L_p. When L_p has a zero at the weight k₀, we present results relating the derivatives at k₀ to Heegner cycles.

Daniel Macias Casillo (King's College London): Values of derivatives of Dirichlet and Hasse-Weil L-functions (13.7.11)

We discuss two different instances of a general underlying aim to generalize statements often studied in the literature involving the values at certain integers of arithmetic L-functions to analogous statements involving the values of higher order derivatives of these L-functions.

In the first instance, we discuss a conjecture concerning the annihilation, as Galois modules, of ideal class groups by explicit elements constructed from the values at s=0 of higher order derivatives of Dirichlet L-functions, in the spirit of a "higher order Stickelberger Theorem". We describe evidence in support of this conjecture, including a full proof in several important cases.

In the second instance, we discuss certain integral congruences between explicit elements constructed from the values at s=1 of higher order derivatives of twisted Hasse-Weil L-functions, in the spirit of a higher order generalization of the refined Birch and Swinnerton-Dyer Conjectures formulated by Mazur and Tate, and describe evidence in support of our predictions.

Patrick Forré (Universität Regensburg): Class field theory for varieties over local fields (22.7.11)

For smooth proper varieties X over a local field, a reciprocity map from a group SK₁(X) to the abelianized fundamental group of X was constructed in work by Bloch, Kato and S. Saito. Saito showed that for a curve the reciprocity map has divisible kernel and a cokernel which can be described by the configuration of the reduction graph. In work by Jannsen and Saito it was shown that there is a similar description for the cokernel in general, and that the kernel is a direct sum of a finite group and a divisible group. We show that the last fact holds in any dimension.

Andreas Nickel (Universität Regensburg): Annihilation of higher K-groups (22.7.11)

Let L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well known Coates-Sinnott conjecture. We discuss the relation to the appropriate special case of the equivariant Tamagawa number conjecture and also provide some non-conjectural evidence for our conjecture.