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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

Date Speaker Title Remarks
4.5.11 Discussion
11.5.11 Vikram Mehta (Tata Institute of Fundamental Research) The Density of the Fundamental Group Scheme
18.5.11 Claudio Pedrini (University of Genoa)
25.5.11 Werner Bley (LMU München) ETNC und elliptische Kurven
1.6.11 Andreas Rosenschon (LMU München) Algebraische Zyklen in Kodimension 2
8.6.11 Stefan Gille (LMU München) Chowmotive und Rost-Nilpotenz
15.6.11 Jan Steffen Müller (Universität Hamburg) Algorithmische Aspekte der Vermutung von Birch und Swinnerton-Dyer
22.6.11 Marco Seveso (Università di Milano) Families of modular forms, p-adic interpolation and p-adic Gross-Zagier type formulas
29.6.11 Ruben Debeerst (Universität Kassel) Die Epsilonkonstantenvermutung
13.7.11 Daniel Macias Casillo (King's College London) Values of derivatives of Dirichlet and Hasse-Weil L-functions
Freitag, 22.7.11 gemeinsames Seminar München-Regensburg in München (Hörsaal B 132)
14:30-15:30 Uhr Patrick Forré (Universität Regensburg) Class field theory for varieties over local fields
16:00-17:00 Uhr Andreas Nickel (Universität Regensburg) Annihilation of higher K-groups
17:15-18:15 Uhr Werner Bley (LMU München) The ETNC for the base change of an elliptic curve
27.7.11 Norbert Hoffmann (FU Berlin) Die essentielle Dimension von Vektorbündeln auf einer Kurve
Freitag, 29.7.11 Yuri Tschinkel (New York University) Rationale Punkte und rationale Kurven auf algebraischen Varietäten im Kolloquium

 


 

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 Fk be its weight k+2 specialization. Up to a trascendental period, the special values of the complex L-functions L(Fk,χ,s) (resp. L(Fk/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 Lp. When Lp 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.