### Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2010/11

 19.10.2010 Vorbesprechung 26.10.2010 Nahid Walji (HU Berlin) The Lang Trotter conjecture on average and congruence class bias 02.11.2010 Anilatmaja Aryasomayajula (HU Berlin) Bounds on canonical Green's function 09.11.2010 Stefan Keil (HU Berlin) The Sato-Tate Conjecture and the L-Function Method Abstract: In the last years Tayler et al. showed that the m-th symmetric power of the L-functions of l-adic Galois-representations of an elliptic curve over QQ are potential automorphic. Therefore they fulfill certain meromorphic properties which are sufficient to use the so called L-function method to prove the Sato-Tate Conjecture (over QQ). We will talk in general about equidistribution and the L-function method and its application (e.g. the prove of Chebotarev's density theorem). We will explain why one can apply the L-function method for the Sato-Tate Conjecture. If there is time we will discuss the exceptional case of CM-curves. 16.11.2010 Remke Kloosterman (HU Berlin) Mordell-Weil groups, Syzygies and Brill-Noether theory Abstract: For a particular class of elliptic threefolds with base P^2 we discuss the relation between the rank of the Mordell-Weil group and the syzygies of the singular locus of the discriminant curve of the elliptic fibration. From this we deduce that for each high rank elliptic threefold we find examples of triples (g,k,n) such that the locus {[C] \in M_g  |   C admits a   g^2_{6k} and the image of C has at least n cusps} has much bigger dimension than expected. 23.11.2010 kein Seminar 30.11.2010 Luis Dieulefait (Universitat de Barcelona) Modularity by propagation: Serre's conjecture and non-solvable base change for GL(2) 07.12.2010 Jose Ignacio Cogolludo (Zaragoza) On the connection between the topology of plane curves, the position of their singular points, and elliptic threefolds 14.12.2010 Brendan Creutz (Univ. Bayreuth) Large Tate-Shafarevich Groups Abstract: For an abelian variety A over a number field k, the Tate-Shafarevich group of A/k parameterizes principal homogeneous spaces for A/k which have points over every completion of k. It is conjectured that this group is finite. However, there are several results in the literature which show that this group can be arbitrarily large. We will discuss some of these and show, in particular, that the p-torsion in the Tate-Shafarevich group of any principally polarized abelian variety over k is unbounded as one ranges over extensions of k of degree O(p). 04.01.2011 kein Seminar 11.01.2011 Jens Funke (Univ. of Durham/BMS) On a theorem of Hirzebruch and Zagier 18.01.2011 Jay Jorgenson (City College New York/BMS) Progress in bounds for Fourier coefficients of modular forms 25.01.2011 Barbara Jung (HU Berlin) On the volume formula of the fundamental domain of the Siegel modular group 01.02.2011 Jürg Kramer (HU Berlin) On a result of Hoffstein-Lockhart 08.02.2011 Lejla Smajlović (Univ. of Sarajevo) On the distribution of the zeros of the derivative of the Selberg zeta function 15.02.2011 Anna-Maria von Pippich (HU Berlin) The Weierstrass $\wp$-function and transcendence questions Freitag 18.03.2011 Alex Bartel (POSTECH, Pohang Univ., Korea) Some applications of integral group representations in number theory Abstract: Certain quotients of regulators of number fields or of abelian varieties can be interpreted as purely representation theoretic invariants. I will introduce a technique for analysing such invariants and will apply it to some questions on the arithmetic of number fields and of elliptic curves. One of the main results will be a curious identity linking the fine integral Galois module structure of certain units of number fields and of Mordell-Weil groups to sizes of class groups and of Tate-Shafarevich groups, respectively. There will be plenty of examples along the way, and also some mystery and food for future thought. The talk will be accessible to graduate students with a very modest background in number theory and ordinary representation theory.