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

## Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2013

Das Forschungsseminar findet dienstags in der Zeit von 13.00 - 15.00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.

 09.04.2013 Jan Bruinier (TU Darmstadt) Heights of Kudla-Rapoport divisors and derivatives of L-functions 16.04.2013 entfällt 23.04.2013 Niels Lindner (HU Berlin) Bertini's theorem for weighted projective space over a finite field Abstract: The classical Bertini theorem states that a general hypersurface in complex projective space is smooth. Given a smooth subvariety X of projective space over a finite field, one can actually calculate the fraction of hypersurfaces whose intersection with X is again smooth. This number can be expressed in terms of the Zeta function of X. The question makes also sense when replacing "projective space" with "weighted projective space" and "smooth" with "quasismooth". However, the nature of weighted projective space raises some new difficulties. 30.04.2013 Giovanni De Gaetano (HU Berlin) Remarks on Determinants of Laplacians on Riemann Surfaces 07.05.2013 Philipp Bannasch (HU Berlin) Thetadivisoren auf elliptischen Modulflächen und ihr Schnittverhalten 14.05.2013 Stefan Keil (HU Berlin) Non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals Abstract: For an elliptic curve (over a number field) it is known that the order of its Tate-Shafarevich group is a square, provided it is finite. In higher dimensions this no longer holds true. We will present work in progress on the classification of all occurring non-square parts of orders of Tate-Shafarevich groups of non-simple abelian surfaces over the rationals. We will prove that only finitely many cases can occur. To be precise only the cardinalities k=1,2,3,5,6,7,10,13,14,26 are possible. So far, for all but the last three cases we are able to show that these cases actually do occur by constructing explicit examples. 21.05.2013 Banafsheh Farang-Hariri (HU Berlin) Local and global theta correspondence Abstract: I will start by describing the local theta correspondence also known as Howe correspondence over a local non-archimedean field. This correspondence relates different class of representations of two reductive groups (G,H) called a dual pair by means of the Weil representation. Then I will talk about the global theta correspondence and it's relation with the local one. I will also explain the link between global theta correspondence and theta series in the theory of automorphic forms. 28.05.2013 Shaoul Zemel (TU Darmstadt) A Gross-Kohnen-Zagier Type Theorem for Higher-Codimensional Heegner Cycles Abstract: The multiplicative Borcherds singular theta lift is a well-known tool for obtaining automorphic forms with known zeros and poles on quotients of orthogonal symmetric spaces. This has been used by Borcherds in order to prove a generalization of the Gross-Kohnen-Zagier Theorem, stating that certain combinations of Heegner points behave, in an appropriate quotient of the Jacobian variety of the modular curve, like the coeffcients of a modular form of weight 3/2. The same holds for certain CM (or Heegner) divisors on Shimura curves. The moduli interpretation of Shimura and modular curves yields universal families (Kuga-Sato varieties) over them, as well as variations of Hodge structures coming from these universal families. In these universal families one defines the CM cycles, which are vertical cycles of codimension larger than 1 in the Kuga-Sato variety. We will show how a variant of the additive lift, which was used by Borcherds in order to extend the Shimura correspondence, can be used in order to prove that the (fundamental cohomology classes of) higher codimensional Heegner cycles become, in certain quotient groups, coefficients of modular forms as well. Explicitly, by taking the $m$th symmetric power of the universal family, we obtain a modular form of the desired weight 3/2+m. 04.06.2013 entfällt (Sommerschule Blossin) 11.06.2013 entfällt (GRK 1800 Intensivkurs) 18.06.2013 Lejla Smajlovic (U Sarajevo) On Maass forms and holomorphic modular forms on certain moonshine groups. Abstract: We present results on analytical and numerical study of Maass forms and holomorphic modular forms on moonshine groups of level N , where N is a squarefree positive integer. We derive "average" Weyl's law for the distribution of discrete eigenvalues of Maass forms from which we deduce the "classical" Weyl's law. The groups corresponding to levels N=5 and N=6 have the same signature; however, our analysis shows that there are infinitely more cusp forms for N=5. Furthermore, we deduce a Kronecker limit formula for parabolic Eisenstein series and express the "Kronecker limit function" as a geometric mean of product of classical eta functions. We also study holomorphic forms non-vanishing at the cusp and discuss the construction of the j-function(s). 25.06.2013 David Ouwehand (HU Berlin) Rigid cohomology at singular points and the computation of zeta functions Abstract: The topic of this talk is the computation of the action of Frobenius on the rigid cohomology of the complement of certain singular projective hypersurfaces over a finite field. Abbott, Kedlaya and Roe have developed a method that solves this problem for smooth hypersurfaces, but for singular hypersurfaces there are still many open questions left. We start by introducing a notion of equivalence of singularities for varieties over finite fields. This notion has the advantage that two equivalent singularities have isomorphic local rigid cohomology. Then we discuss a method for dealing with varieties having weighted homogeneous singularities by using an idea of Dimca. This is where the understanding of the local cohomology at singular points plays a key role. 02.07.2013 Barbara Jung (HU Berlin) The star product of Green currents on the Siegel modular variety of degree two Abstract: To obtain the arithmetic degree of the Hodge bundle on (a compactification of) the Siegel modular variety A_2 of degree two, the fourfold intersection product of the bundle of modular forms equipped with the Petersson metric has to be computed. This leads to an integral over the regularized star product of corresponding Green currents on A_2. Choosing appropriate currents, we obtain a decomposition in computable integrals over cycles on A_2. We will carry out this decomposition, evaluate the integrals and compare with the expected value. 09.07.2013 Adrian Vasiu (U Binghamton) Generalized Serre-Tate Ordinary Theory Abstract: In the late 60's, Serre and Tate developed an ordinary theory for abelian varieties over a perfect field of positive characteristic. The ordinary locus corresponds to the generic, dense, open stratum of the Newton polygon stratification of the moduli spaces of principally polarized abelian varieties of a fixed dimension. In this talk, we report on a generalized Serre- Tate ordinary theory that involves Shimura-ordinariness and Uni-ordinariness, in both abstract and geometric contexts. In particular, geometric applications to the study of special fibres of integral canonical models of Shimura varieties of Hodge type and the Shimura-ordinary loci will be presented.
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Wintersemester 2007/08
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