Humboldt Universität zu Berlin
Mathem.Naturwissenschaftliche Fakultät
Institut für Mathematik
Sommersemester 2014
Das Forschungsseminar findet dienstags in der Zeit von 13.15  15.00 Uhr in der Rudower Chaussee 25, 12489 BerlinAdlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.
15.04.2014  Vorbesprechung 
22.04.2014  Tonghai Yang (University of Wisconsin, zzt. MPI Bonn) 
The story of $j$ and generalizations Abstract: The classical modular invariant $j$ has a long history. For example, it was long known that $j(\frac{D +\sqrt D}2)$ is an algebraic integer generating the Hilbert class field of the imaginary quadratic field $Q(\sqrt D)$. As early as 1920's, Berwick observed that the norm of the difference of some singular moduli, like $j(\frac{163+\sqrt{163}}2) j(i)$, although very big, has a very small and interesting prime factorization. In 1985, Gross and Zagier confirmed this guess and gave a beautiful factorization form in general. In 1990s, Borcherds, in attempting to prove the celebrated Moonshine conjecture, discovered and proved a beautiful and surprising product formula for the modular function $j(z_1) j(z_2)$. His idea of 2nd proof, the regularized theta lifting, can be used to prove GrossZagier's formula easily. Moreoever, the method is very soft and can be extended to give direct link between the central derivative of some Lseries and the height pairing on some Shimura varieties of orthogonal/unitary type. In this talk, I will mainly focus on the proof of GrossZagier formula using regularized theta lifting (less notation and general concepts) after some background review. In the last 2030 minutes, I will explain the extension. 

29.04.2014  kein Seminar 
06.05.2014  Yuri Bilu (Université de Bordeaux) 
Mini Course: The Subspace Theorem in Diophantine Analysis Part I Abstract: I will explain the statement of the Subspace Theorem of Schmidt and Schlickewei and will show some of its applications. In particular, I will prove the AdamcszewkiBugeaud theorem on transcendence of automatic numbers, and the theorem of CorvajaZannierLevinAutissier on the nondensity of integral points on algebraic surfaces. 

13.05.2014  Yuri Bilu (Université de Bordeaux) 
Mini Course: The Subspace Theorem in Diophantine Analysis Part II  
Abstract: I will speak on the work of Corvaja, Zannier and others, on applying the Subspace Theorem to integral points on curves and surfaces.  
20.05.2014  Remke Nanne Kloosterman (HU Berlin) 
The Hilbert function of the singular locus of hypersurfaces  
Abstract: Given a zero dimensional scheme $X$ in $P^n$, it is hard to determine if $X$ occurs as the singular locus of a degree d hypersurface. In this talk we use some results on the topology of singular hypersurfaces to obtain restrictions on the Hilbert function of the singular locus of a degree d hypersurface with isolated singularities. This result has several corollaries: It enables us to determine the MordellWeil rank of several isotrivial fibrations of abelian varieties. Moreover, it enables us to give constructions of SeveriEnriques varieties with dimension bigger than expected. For the final application let $p_1,\dots,p_t$ be points in $P^n$ and$ $m an integer. We give a nontrivial lower bound for the degree of a hypersurface in $P^n$ with $m$fold points at the $p_i$ and no further singularities.  
27.05.2014  Zavosh AmirKhosravi (HU Berlin) 
Double coset spaces for the compact unitary groups  
Abstract: I will speak about B. Gross's construction of a definite Shimura curve, its special points and height pairing, and the GrossZagiertype formula to which this leads. I will then describe an ongoing project to carry out a similar construction for the compact unitary group U(n), replacing special points by certain special cycles, and defining a corresponding notion of heights. One expects to obtain a generating series that's an automorphic form on U(r,r) with coefficients in Chow groups.  
03.06.2014  Siddarth Sankaran (Universität Bonn) 
Title: Towards an arithmetic SiegelWeil formula  
Abstract: The SiegelWeil formula, first discovered by Siegel in 1951, is a classical theorem that equates the integral of a theta function with a special value of an Eisenstein series. A beautiful series of papers by Kudla and Millson from the late 80's casts this theorem a geometric light, in terms of certain 'special cycles' on Hermitian symmetric domains and their quotients. Since then, evidence has emerged for a deeper arithmetic significance of this geometric interpretation; in many cases, there are deep and surprising connections between integral models of these cycles on the one hand, and derivatives of Eisenstein series on the other. In this talk, I will introduce this circle of ideas, which has come to be known as Kudla's programme, and in particular focus on recent developments in the context of unitary Shimura varieties.  
10.06.2014  kein Seminar 
17.06.2014  Miguel Grados Fukuda (HU Berlin) 
Zeta functions of ray ideal classes and applications to Arakelov geometry  
Abstract: Pursuing asymptotic formulas for the selfintersection number of the relative dualizing sheaf of a modular curve, one encounters zeta functions associated to congruence subgroups. In this talk, we identify such zeta functions and those attached to ray ideal classes of real quadratic fields. The purpose of this identification is to obtain a residue formula at s=1 for the former class of zeta functions. The special case of $\Gamma(N)$ is worked out in detail. If time allows, an idelic interpretation of this approach will be presented.  
24.06.2014  Slawomir Cynk (Jagiellonian University Krakow, z.Zt. Universität Mainz) 
PicardFuchs operators for one parameter families of CalabiYau threefolds  
Abstract: I will present some methods of computation of PicardFuchs operators for one parameter families of CalabiYau threefolds. The motivation comes from the Mirror Symmetry and an attempt to classify CalabiYau threefolds with the Picard number equal 1. I will discuss examples with special properties: families without points of maximal unipotent monodromy (MUM) and families with several MUM points. This is joint research with D. van Straten (Mainz).  
01.07.2014  Ana Maria Botero (HU Berlin) 
The singularities of the invariant metric on the line bundle of Jacobi forms on the universal elliptic curve  
Abstract: A theorem by Mumford implies that an automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety acquiring only logarithmic singularities. This result is the key point of being able to compute arithmetic intersection numbers from these line bundles. It is then natural to ask, whether this result extends to mixed Shimura varieties. In this talk, we examine the case of the sheaf of Jacobi forms on the universal elliptic curve. We see that Mumford's theorem cannot be applied here since a new kind of singularities appear. However, we show that a natural extension in this case is a so called bdivisor. This extension is meaningful because it satisfies ChernWeil theory and a HilbertSamuel type formula. This is work by J. I. Burgos, J. Kramer and U. Kühn.  
08.07.2014  Giovanni de Gaetano (HU Berlin) 
A metric degeneration approach to a special case of the arithmetic Riemann  Roch theorem  
Abstract: In 2010 G. Freixas proved an arithmetic Riemann – Roch theorem for arbitrary powers of the bundle of cusp forms on modular curves equipped with logsingular metrics; his approach heavily relied on the properties of the moduli space of pointed Riemann surfaces. The goal of the talk is to introduce a different method to prove this theorem, namely the metric degeneration process, which does not require the existence of such a moduli space. Partial results and open problems of such a program will be widely discussed.  
15.07.2014  David Ouwehand (HU Berlin) 
Local rigid cohomology of weighted homogeneous singularities  
Abstract: In 2005 Abbott, Kedlaya and Roe have given a method to compute the rigid cohomology of a smooth hypersurface. The goal of this talk is to explain how this method can be applied to the computation of the local rigid cohomology of weighted homogeneous singularities. 
Wintersemester 2007/08
