### Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2007/08

 30.10.2007 Christian Christensen (Univ. Dortmund, z.Z. HU Berlin) Der relative Satz von Schanuel 06.11.2007 Gabor Wiese (Univ. Duisburg-Essen, Campus Essen) Zur Erzeugung von Koeffizientenkörpern von Neuformen durch einen einzigen Hecke-Eigenwert 20.11.2007 Aleksander Momot (ETH Zürich) Toroidally compactified ball quotient surfaces in small Kodaira dimension Abstract: As well as Hilbert modular surfaces, compact and compactified quotients $X = \overline{\Gamma \setminus \mathbf{B}}$ of the open complex unit-ball $\mathbf{B} \subset \mathbb{C}^2$ serve as the two-dimensional analog of modular curves. They are thus intimately connected with modular problems, but also provide interesting and extremal examples in the geography of surfaces.\\ While Holzapfel has classified such surfaces for the case that they are defined by Picard modular ball-lattices $\Gamma = \mathbf{PU}(2,1; \mathfrak{a})$, $\mathfrak{a}$ an order in an imaginary quadratic number field, I aim a classification avoiding arithmetic conditions on Gamma. In the main part of the talk I will sketch the classification for kod(X)<= 0, q(X)> 0. 27.11.2007 Walter Gubler (Univ. Dortmund, z.Z. HU Berlin) Tropische analytische Geometrie und die Bogomolov-Vermutung 04.12.2007 Maria Petkova (HU Berlin) Ball quotients curves with application in the coding theory 08.01.2008 Jay Jorgenson (CCNY) Zeta functions and determinants on discrete tori Abstract: For any integer m, we let mZ\Z denote a discrete circle, and we define a discrete torus to be a product of a finite number of discrete circles. Associated to the combinatorial Laplacian, one has a finite set of eigenvalues from which one can form the determinant, namely the product of the eigenvalues. We prove, under reasonably general assumptions, an asymptotic expansion of the determinant as the parameters m in the discrete circles tend to infinity. Specifically, we establish a "lead term" which solely involves the parameters of the discrete circles, and a "second order term" which is a modular form associated to the Kronecker limit problem for Epstein zeta functions. We show that the modular form is that which is obtained when definining the regularized determinant of the Laplacian on real tori, thus establishing a new connection between determinants and zeta regularized determinants, as well as discrete and real tori. 15.01.2008 Amir Dzambic (Univ. Frankfurt a. Main) Geometrie und Arithmetik falscher projektiver Ebenen 22.01.2008 Giancarlo Urzua (Univ. of Michigan) Arrangements of curves and algebraic surfaces Abstract: We show a close relation between Chern and log Chern numbers of complex algebraic surfaces. In few words, given a log surface (Y,D) of a certain type, we prove that there exist smooth projective surfaces X with Chern ratio arbitrarily close to the log Chern ratio of (Y,D). The method is a "random" p-th root cover which exploits a large scale behavior of Dedekind sums and negative-regular continued fractions. We emphasize that the "random" hypothesis is necessary for this limit result. For certain divisors D, this construction controls the irregularity and/or the topological fundamental group of the new surfaces X. For example, we show how to obtain simply connected smooth projective surfaces, which come from the dual Hesse arrangement, with Chern ratio arbitrarily close to 8/3. In addition, by means of the Hirzebruch inequality for complex line arrangements, we show that this is the (unique) best result (closer to the Miyaoka-Yau bound 3) for complex line arrangements. 29.01.2008 Sascha Orlik (Univ. Leipzig) Äquivariante Vektorbündel auf Drinfelds Halbraum 05.02.2008 Jose Ignacio Burgos Gil (Univ. of Barcelona) The arithmetic Riemann Roch theorem for closed immersions 12.02.2008 Emmanuel Ullmo (Universite Paris-Sud, Orsay) Rational points on Shimura varieties