The Bielefeld Arithmetic Geometry Seminar 2016 (summer term)

Welcome to the website of the 7th edition of the Bielefeld arithmetic geometry seminar! Below you will find a list of the upcoming talks, hover with your mouse over abstract to see the abstract. The seminar is related to projects B5 and C7.
The seminar is supported by the SFB CRC 701: Spectral Structures and Topological Methods in Mathematics.

Talks

Wednesday 27 April, 2016
15:00	U2-135	Djordje Milićević (Bryn Mawr College) On moments of twisted $L$-functions [Abstract] In addition to providing statistical and intrinsic information about the underlying family of automorphic forms, asymptotic formulas with a power saving for moments of central values of $L$-functions are an essential ingredient in analytic approaches to questions of arithmetic importance such as upper bounds, nonvanishing, or extreme values. In this talk, I will present recent asymptotic formulas for moments in several families of twisted $L$-functions with all primitive characters modulo $q$, with a power saving in $q$, as well as a sampling of applications. We use a variety of methods, using the full power of spectral theory of $GL(2)$ automorphic forms to treat a possibly highly unbalanced shifted convolution problem, as well as arithmetic and algebro-geometric techniques (including $q$-Weyl differencing, Burgess-Karatsuba method, Riemann Hypothesis for curves over finite fields, and independence of Kloosterman sheaves) to prove large-sieve-type and estimates on bilinear forms in Kloosterman sums in critical ranges. This is joint work with Blomer, Fouvry, Kowalski, Michel, and Sawin.
16:15	U2-135	Sören Kleine (Munich) Local behaviour of Iwasawa invariants [Abstract] Let $K$ be a number field. A $\mathbb{Z}_p$-extension of $K$ is a normal extension $L$ of $K$ such that $Gal(L/K)$ is isomorphic to the additive group $\mathbb{Z}_p$ of $p$-adic integers. Iwasawa has shown that the growth of the $p$-class numbers of the intermediate fields in such a $\mathbb{Z}_p$-extension can be described asymptotically in terms of three invariants, which nowadays are called the 'Iwasawa invariants' attached to the extension $L/K$. In this talk, we will compare the Iwasawa invariants of different $\mathbb{Z}_p$-extensions of some fixed field $K$. The main result provides explicit local upper bounds of these invariants with respect to a suitable topology on the set of all $\mathbb{Z}_p$-extensions of $K$. This substantially exploits the Galois module structure of the ideal class groups of the corresponding intermediate fields. The main ingredient in the proof is a generalisation of a theorem of Fukuda concerning the stabilisation of certain quotients of the ideal class groups.
Wednesday 11 May, 2016
15:00	U2-135	Judith Ludwig (Bonn) $p$-adic automorphic forms and endoscopy [Abstract] Let $f \in S_{k}(\Gamma_1(N))$ be a modular eigenform that has CM by an imaginary quadratic field in which $p$, a fixed prime, splits. Such a form $f$ gives rise to a point $x$ of critical slope on the Coleman-Mazur eigencurve of tame level $N$. It is well known that the space $\mathrm{CM}^{\dagger}_k(\Gamma_1(N)\cap \Gamma_0(p))_{(x)}$ of overconvergent generalized eigenforms for the system of Hecke eigenvalues determined by $x$ contains a non-classical form. In this talk we study the analogous situation for the group $\mathrm{SL}_2$. After recalling some background, I will explain how to use the theory of endoscopy and the geometry of eigenvarieties to show the existence of a non-classical eigenform in the corresponding space.
16:15	U2-135	Francesc Fité (Duisburg-Essen) Fields of definition of CM elliptic $k$-curves with an application [Abstract] Let $A$ be an abelian variety defined over a number field $k$ that is isogenous over $\overline{\mathbb Q}$ to the power of an elliptic curve $E$. If $E$ does not have CM, by results of Ribet and Elkies concerning fields of definition of $k$-curves, $E$ is isogenous to an elliptic curve defined over a polyquadratic extension of $k$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find some applications of this analysis. We show that there are at most 51 possibilities for the $\mathbb{Q}$-algebra of endomorphisms of an abelian surface over $\mathbb{Q}$ that is $\overline{\mathbb{Q}}$-isogenous to the square of an elliptic curve. As a consequence, 18 of the 34 possible Sato-Tate groups of abelian surfaces over $\mathbb{Q}$ only occur among at most 51 $\overline{\mathbb{Q}}$-isogeny classes of abelian surfaces. We also prove the existence of a number field over which abelian surfaces can be found realizing each of the possible Sato-Tate groups of abelian surfaces over number fields. This is a joint work with Xevi Guitart.
Wednesday 25 May, 2016
14:30	U2-135	Valentin Blomer (Göttingen) Eigenfunctions on arithmetic manifolds [Abstract] TBA
Wednesday 22 June, 2016
15:00	U2-135	Danylo Radchenko (Bonn) Higher cross-ratios and functional equations for polylogarithms [Abstract] A well-known conjecture of Zagier states that the special value of a Dedekind zeta function of a number field at an integer $m>1$ can be expressed in terms of the $m$-th polylogarithm function. This conjecture remains widely open for all $m>3$. A general strategy for proving it was outlined by Goncharov, and the main ingredient involves constructing certain higher-dimensional generalizations of the classical cross-ratio. In this talk I will give a general definition of higher cross-ratios, show how they can be used to construct interesting functional equations for polylogarithms, and report on the progress towards proving Zagier's conjecture in the case $m=4$.

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The Bielefeld Arithmetic Geometry Seminar 2016 (summer term)

Talks

Wednesday

27 April, 2016

Wednesday

11 May, 2016

Wednesday

25 May, 2016

Wednesday

22 June, 2016