Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2009/10

13.10.2009 Vorbesprechung
20.10.2009  Kira Samol (Univ. Mainz)
Dwork congruences and reflexive polytopes
27.10.2009 Christian Schön (HU Berlin)
Some \Gamma_1(N) modular forms and their connection to the Weierstrass \wp-function
03.11.2009 Judith Ludwig (Univ. Tübingen)
Parabolic cohomology and rational periods
10.11.2009 Mark Blume (HU Berlin)
Losev-Manin moduli spaces and toric varieties associated with root systems
17.11.2009 Peter Bruin (Univ. Leiden)
Arakelov theory and height bounds
Abstract: In the work of Bas Edixhoven and others on computing two-dimensional Galois representations associated to modular forms over finite fields, part of the output of the algorithm is a certain polynomial with rational coefficients that is approximated numerically. Arakelov's intersection theory on arithmetic surfaces is applied to modular curves in order to bound the heights of the coefficients of this polynomial. I will explain the connection between Arakelov theory and heights, indicate what quantities need to be estimated, and give methods for doing this that lead to explicit height bounds.
24.11.2009 Amaury Thuillier (Lyon 1)
Non-Archimedean analytic geometry and Arakelov theory
Abstract: Arakelov geometry on an algebraic variety X over Q usually combines intersection theory on a model of X over the integers with analysis on the corresponding complex analytic variety. Relying on Berkovich's approach of p-adic analytic geometry, it is possible to replace integral models by (real) analysis at each non-Archimedean place of Q. This extension of Arakelov geometry is particulary suitable for the study of canonical heights, or to formulate and prove p-adic equidistribution theorems. I will explain this without assuming a specific knowledge of Berkovich theory.
01.12.2009 Bisi Agboola (UCSB z.Zt. HU Berlin)
Restricted Selmer groups and special values of p-adic L-functions
Abstract: I shall discuss conjectures of Birch and Swinnerton-Dyer type involving special values of the Katz 2-variable p-adic L-function that lie outsied the range of p-adic interpolation.
05.01.2010 Ananyo Dan (HU Berlin)
Classifying the biggest components of the Noether-Lefschetz locus
12.01.2010 Ricardo Menares (EPFL Lausanne)
Functoriality in the Arakelov geometry of arithmetic surfaces. Applications to Hecke operators on modular curves
Abstract: In the context of arithmetic surfaces, J.-B. Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pullback and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are selfadjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due to U. Kuehn we compute these invariants in terms of special values of L series.
19.01.2010 Fredrik Strömberg (TU Darmstadt)
On newforms and multiplicity of the spectrum for Gamma_0(9)
Abstract: Through numerical investigations it was discovered (by two independent groups of researchers) some years ago that the spectrum of the Laplace-Beltrami operator on $\Gamma_0(9)$ possessed a peculiar feature. Namely, that there did not seem to be any eigenvalues with multiplicity one, that is, even the newforms (in the sense of Atkin-Lehner extended to non-holomorphic forms) appeared to be degenerate. The underlying phenomenon providing this multiplicity became only recently clear. I will give an overview of the proof of a precise version of this observation. The proof involves a mix of spectral theory, in terms of the Selberg trace formula, together with Hecke operators and symmetries of a (slightly overlooked) congruence group of level 3, $\Gamma^3$. I will also discuss how this relates to some of the multiplicity one results for automorphic representations.
26.01.2010 Stephen Meagher (Univ. Freiburg)
Equations defining isogeny classes of ordinary Abelian varieties
Abstract: The moduli space of Abelian varieties can be described in terms of equations which derive from relations holding between the theta null values of Abelian varieties. In characteristic p one can look at the action of the Frobenius and Vershiebung morphisms on these theta nulls when the variety is ordinary. From the work of Tate it is known that the characteristic polynomial of an iterate of the Frobenius morphism describes the isogeny type of an Abelian variety. Using this we describe relations between the level p theta nulls in an isogeny class of ordinary Abelian varieties. This is joint work with Robert Carls.
02.02.2010 Barbara Jung (Univ. Mainz)
A version of Luna's theorem for symplectic varieties
Abstract: In 1973, Domingo Luna proved the existence of an etale slice for certain group actions on varieties: He showed that the algebraic quotient X//G of a variety X by a reductive group G can etale locally be described by the action of the stabilizer G_x of a point x on a subvariety S of X. A symplectic variety carries the additional structure of a non degenerate 2-form. To obtain a suitable quotient in the category of symplectic varieties, one has to pass to a certain subvariety Z_X first before taking the algebraic quotient. For this symplectic quotient, Luna's theorem had to be modified in such a way that we found a symplectic subvariety S of X such that there is an etale morphism from the symplectic quotient Z_S//G_x of S by G_x to the symplectic quotient ZX//G of X by G.
09.02.2010Chad Schoen (Duke University, z.Zt. MPIM Bonn)
A numerical test of the generalized Birch and Swinnerton-Dyer Conjecture
Abstract: The generalized Birch and Swinnerton-Dyer Conjecture in its crudest form asserts that the rank of a Chow group is equal to the order of vanishing of an L-function at the center of the critical strip. We discuss joint work with Jaap Top and Joe Buhler in which the conjecture was put to a modest test.