The Bielefeld Arithmetic Geometry Seminar 2013 (summer term)

This is the website of the arithmetic geometry seminar of the arithmetic geometry group of the university of Bielefeld, related to projects B5 and C7.
The seminar is supported by the SFB CRC 701: Spectral Structures and Topological Methods in Mathematics.

Talks

Wednesday April 10, 2013
16:00	V5-227	Christian Kappen (University of Essen) Abelian varieties over number fields, tame ramification and big Galois image [Abstract] In this talk, I will discuss a variant of the potential tame inverse Galois problem, with a view towards abelian varieties. More specifically, I will show that if $K$ is a number field and if $n$ is a positive integer, then for all prime numbers $\ell$ outside a finite exceptional set (depending on $K$ and $n$), there exists a finite field extension $F/K$ which is unramified above $\ell$, together with a principally polarized abelian $F$-variety $A$ such that the $\ell$-torsion representation attached to $A$ is surjective onto the general symplectic group and everywhere tamely ramified. This is joint work with Sara Arias-de-Reyna. Our proof uses moduli spaces of abelian varieties, non-archimedean analytic geometry and results of Moret-Bailly on the existence of global points.
Wednesday April 17, 2013
16:00	V4-116	Malte Witte (University of Heidelberg) Noncommutative Iwasawa Main Conjecture in the Function Field Case [Abstract] The main conjecture of Iwasawa theory describes a connection between the module structure of e.g. the Selmer group of an elliptic curve over a $p$-adic Lie extension of the ground field as module over the associated Iwasawa algebra and a noncommutative $p$-adic $L$-function, which interpolates special values of classical $L$-functions. In my talk I will consider the analogue of the noncommutative main conjecture for function fields of characteristic different from $p$. In this case, there exists a complete proof of the main conjecture.
Wednesday May 15, 2013
15:00	V5-227	Guido Kings (University of Regensburg) Special values of p-adic L-functions for modular forms [Abstract] The Beilinson conjecture predicts that the special values of the $L$-function of a motive can be interpreted by special elements in algebraic $K$-theory and their image in Deligne cohomology via the Beilinson regulator map. For modular forms Beilinson and Scholl constructed such elements and verified Beilinson's conjecture for them. Perrin-Riou has conjectured that the same elements interpret also the values of the $p$-adic $L$-function, if one considers their image in syntomic cohomology via the syntomic regulator. In this lecture we explain how one can compute the syntomic regulator of the $K$-theory elements for modular forms in terms of the $p$-adic convolution measure introduced by Panchishkin. As a consequence Perrin-Riou's conjecture can be verified for modular forms.
Wednesday May 29, 2013
16:00	V4-116	Thanasis Bouganis (University of Heidelberg) On special values of Siegel modular forms [Abstract] In this talk I will discuss various algebraicity results for special values of $L$-functions attached to Siegel modular forms. There are in general two methods for studying these $L$-values. The so-called Doubling Method and the Rankin-Selberg Method. I will start by explaining these methods and then building on results of Shimura (and also of Sturm, Harris, Panchishkin, Boecherer-Schmidt). I will explain in some cases various reciprocity laws of the action of the absolute Galois group on these special values. If time allows, I will also explain similarities and differences with the case of hermitian modular forms i.e. automorphic forms attached to unitary groups.
Wednesday June 26, 2013
16:00	V5-227	Andre Chatzistamatiou (University of Essen) De Rham-Witt Cohomology [Abstract] Let $X$ be a smooth projective $R$-scheme, and let $R$ be an \'etale $\mathbb{Z}$-algebra. As constructed by Hesselholt, we have the absolute big de Rham-Witt complex $W\Omega_X$ of $X$ at our disposal. There is also a relative version $W\Omega_{X/\mathbb{Z}}$ that is characterized by the vanishing of the positive degree part in the case $X = \text{Spec}(\mathbb{Z})$. We study the hypercohomology of the relative (big) de Rham-Witt complex of $X$. We show that it is a projective module over the ring of (big) Witt vectors of $R$, provided that the de Rham cohomology is torsion-free. In addition, we establish a Poincar\'e duality theorem. Our results rely on an explicit description of the relative de Rham-Witt complex of a smooth $\lambda$-ring, which may be of independent interest.
Wednesday July 10, 2013
16:30	V5-227	Christian Liedtke (Technical university of Muenchen) Supersingular K3 surfaces are unirational [Abstract] We show that supersingular K3 surfaces are related by purely inseparable isogenies. As an application, we deduce that they are unirational, which confirms conjectures of Artin, Rudakov, Shafarevich, and Shioda. The main ingredient in the proof is to use the formal Brauer group of a Jacobian elliptically fibered supersingular K3 surface to construct a family of "moving torsors" under this fibration that eventually relates supersingular K3 surfaces of different Artin invariants by purely inseparable isogenies.
Tuesday July 16, 2013
16:15	V5-227	Jenna Rajchgot (University of Michigan) Doubly universal Groebner bases [Abstract] A universal Groebner basis of an ideal in a polynomial ring is a finite set of polynomials which is a (non-reduced, non-minimal) Groebner basis for every monomial order. In this talk, I'll explain a way to generalize this notion from an ideal in a polynomial ring to an ideal sheaf defining a universal family. I'll then discuss how one can use this ``doubly universal'' Groebner basis to study torus orbits in Hilbert schemes. This is joint work with Mathias Lederer.
Wednesday July 17, 2013
16:00	V5-227	Jasmin Matz (University of Bonn) An upper bound for global coefficients in Arthur's trace formula for $GL(n)$ [Abstract] We will discuss an explicit upper bound for the so-called global coefficients appearing on the geometric side of Arthur's trace formula in the case of $GL(n)$. These coefficients are in general left unspecified, but a better understanding of them is essential for applications, and I will explain how the upper bound might be useful in proving a Weyl's law for Hecke operators on $GL(n)$.
Thursday July 18, 2013
17:15	V5-227	Adrian Vasiu (University of Binghamton) On the Mumford-Tate Conjecture for abelian varieties over number fields [Abstract] We report on the prior work done and on the present status of this conjecture.

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Seminars

Conference

The Bielefeld Arithmetic Geometry Seminar 2013 (summer term)

Talks

Wednesday

April 10, 2013

Wednesday

April 17, 2013

Wednesday

May 15, 2013

Wednesday

May 29, 2013

Wednesday

June 26, 2013

Wednesday

July 10, 2013

Tuesday

July 16, 2013

Wednesday

July 17, 2013

Thursday

July 18, 2013