### Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2008/09

 14.10.2008 Vorbesprechung 21.10.2008 kein Seminar 28.10.2008 Jürg Kramer (HU Berlin) Estimating Green's Functions I 04.11.2008 Anna von Pippich (HU Berlin) Elliptic and hyperbolic degeneration 11.11.2008 kein Seminar 18.11.2008 Maria Petkova (HU Berlin) Shimura curve computation 25.11.2008 Anilatmaja Aryasomayajula (HU Berlin) Hyperbolic and canonical metrics 02.12.2008 Rolf-Peter Holzapfel (HU Berlin) Picard modular del Pezzo surfaces 09.12.2008 Philipp Habegger (ETH Zürich) Points with multiplicative dependent coordinates on curves Abstract: The Mordell Conjecture states that a smooth projective curve of genus at least 2 has only finitely many points defined over a number field. This is now a theorem of Faltings; another proof using completely different ideas was found by Vojta. His approach can be encapsulated neatly in a single height inequality. Remond found a more uniform version of this inequality and also applications to abelian varieties in the context of the Zilber-Pink Conjecture governing the intersection of a variety with the union of certain algebraic subgroups. Maurin lated built up on Remond's result in the toric case and showed that the curve contained in the algebraic torus of dimension 6 parametrized by (2,3,5,t,1-t,1+t) contains only finitely many points whose coordinates satisfy two independent multiplicative relations. Based ultimately on Vojta's method, Maurin's Theorem is ineffective: it does not allow us to determine the t's. We propose a different approach which circumvents Vojta's method and which is in principle effective in the toric setting. 16.12.2008 kein Seminar 06.01.2009 K. Künnemann (Univ. Regensburg) Line bundles with connections on projective varieties over function fields and number fields Abstract: We report about joint work with with Jean-Benoit Bost (Orsay). Consider a hermitian line bundle on a smooth, projective variety over a number field. The arithmetic Atiyah class of the hermitian line vanishes by definition if and only if the unitary connection on the hermitian line bundle is already defined over the number field. We show that this can happen only if the class of the line bundle is torsion. This problem may be translated into a concrete problem of diophantine geometry, concerning rational points of the universal vector extension of the Picard variety. We investigate this problem, which was already considered and solved in some cases by Bertrand, by using a classical transcendence result of Schneider-Lang. We also consider a geometric analog of our arithmetic situation, namely a smooth, projective variety which is fibered on a curve defined over some field of characteristic zero. To any line bundle on the variety is attached its Atiyah class relative to the base curve. We describe precisely when this relative class vanishes. In particular, when the fixed part of the relative Picard variety is trivial, this holds only when the restriction of the line bundle to the generic fiber of the fibration is a torsion line bundle. 13.01.2009 Rolf-Peter Holzapfel (HU Berlin) Picard modular del Pezzo surfaces, II 20.01.2009 Jay Jorgenson (CCNY) Sup-norm bounds for weight k automorphic forms 27.01.2009 Jean-Benoit Bost (Univ. Paris-Sud, Orsay) Unitary integrable connections defined over number fields 03.02.2009 Amir Dzambic (Univ. Frankfurt am Main) Hyperbolic 3-manifolds with maximal automorphism group 10.02.2009 Aaron Greicius (HU Berlin) Elliptic curves with surjective adelic Galois representation Abstract: Let E/K be an elliptic curve over a number field K. Let G_K be the absolute Galois group of K. The action of G_K on the torsion points of E gives rise to an \emph{adelic} Galois representation $$\rho:G_K\rightarrow Aut(E_{tor}(\overline{K}))\simeq \GL_2(\hat{\mathbb Z}).$$ In 1972 Serre proved that the image of \rho is open, hence of finite index, when E/K is non-CM. The question naturally arises then whether this index is ever equal to 1. In other words, is it possible for \rho to be surjective? By examining the maximal closed subgroups of $GL_2(\hat{\mathbbZ})$, we come up with simple necessary and sufficient conditions for this to be the case. This allows us to find examples of number fields $K\ne\mathbb{Q}$ and elliptic curves E/K for which \rho is indeed surjective.