Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

## Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2014

Das Forschungsseminar findet dienstags in der Zeit von 13.15 - 15.00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, 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 Gross-Zagier's formula easily. Moreoever, the method is very soft and can be extended to give direct link between the central derivative of some L-series and the height pairing on some Shimura varieties of orthogonal/unitary type. In this talk, I will mainly focus on the proof of Gross-Zagier formula using regularized theta lifting (less notation and general concepts) after some background review. In the last 20-30 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 Adamcszewki-Bugeaud theorem on transcendence of automatic numbers, and the theorem of Corvaja-Zannier-Levin-Autissier on the non-density 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 Mordell-Weil rank of several isotrivial fibrations of abelian varieties. Moreover, it enables us to give constructions of Severi-Enriques varieties with dimension bigger than expected. For the final application let $p_1,\dots,p_t$ be points in $P^n$ andm an integer. We give a non-trivial 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 Amir-Khosravi (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 Gross-Zagier-type 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 Siegel-Weil formula Abstract: The Siegel-Weil 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 self-intersection 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) Picard-Fuchs operators for one parameter families of Calabi-Yau threefolds Abstract: I will present some methods of computation of Picard-Fuchs operators for one parameter families of Calabi-Yau threefolds. The motivation comes from the Mirror Symmetry and an attempt to classify Calabi-Yau 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 b-divisor. This extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel 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 log-singular 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.

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Wintersemester 2007/08
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