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

Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2018

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.

24.04.2018 Ulf Kühn (Uni Hamburg)
Title: Multiple q-zeta values and period polynomials
Abstract: We present a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we will state a dimension conjecture for the spaces of their weight- and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst- Kreimer for multiple zeta values.
08.05.2018 Emre Sertöz (MPI Leipzig)
Title: Computing and using periods of hypersurfaces
Abstract: The periods of a complex projective manifold X are complex numbers, typically expressed as integrals, which give an explicit representation of the Hodge structure on the cohomology of X. Although they provide great insight, periods are often very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will give a different algorithm which can compute the periods of any smooth projective hypersurface and can do so with much higher precision. As an application, we will demonstrate how to reliably guess the Picard rank of quartic K3 surfaces and the Hodge rank of cubic fourfolds from their periods.
15.05.2018 Robin de Jong (Uni Leiden)
Title: To be announced
Abstract: Tba
22.05.2018 Roberto Laface (TU München)
Title: Picard numbers of abelian varieties in all characteristics
Abstract: I will talk about the Picard numbers of abelian varieties over C (joint work with Klaus Hulek) and over fields of positive characteristic. After providing an algorithm for computing the Picard number, we show that the set Rg of Picard numbers of g-dimensional abelian varieties is not complete if g ≥ 2, that is there esist gaps in the sequence of Picard numbers seen as a sequence of integers. We will also study which Picard numbers can or cannot occur, and we will deduce structure results for abelian varieties with large Picard number. In characteristic zero we are able to give a complete and satisfactory description of the overall picture, while in positive characteristic there are several pathologies and open questions yet to be answered.
12.06.2018 Giuseppe Ancona (Uni Strasbourg)
Title: On the standard conjecture of Hodge type for abelian fourfolds
Abstract: Let S be a surface and V be the Q-vector space of divisors on S modulo numerical equivalence. The intersection product defines a non degenerate quadratic form on V. We know since the Thirties that it is of signature (1,d-1), where d is the dimension of V. In the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is an easy consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable. Moreover, using a classical product formula on quadratic forms, the p-adic result will give us non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.
17.07.2018 Javier Fresán (École Polytechnique)
Title: To be announced
Abstract: Tba


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