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: The Néron-Tate heights of cycles on abelian varieties
Abstract: Given a polarized abelian variety A over a number field and an effective cycle Z on A, there is naturally attached to Z its so-called Néron-Tate height. The Néron-Tate height is always non-negative, and behaves well with respect to multiplication-by-N on A. Its good properties have led and still lead to applications in number theory. We discuss formulas for the Néron-Tate heights of some explicit cycles. First we focus on the tautological cycles on jacobians, where we find a new proof of the Bogomolov conjecture for curves. Then we focus on the symmetric theta divisors on a general principally polarized abelian variety. Here we find an explicit relation with the Faltings height. The latter part is based on joint work with Farbod Shokrieh.
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.
29.05.2018 Barbara Jung (HU Berlin)
Title: The arithmetic self intersection number of the Hodge bundle on A2
Abstract: The Hodge line bundle ω on the moduli stack A2, metrized by the L2-metric, can be identified with the bundle of Siegel modular forms, metrized by the logarithmically singular Petersson metric. Intersection theory for line bundles on arithmetic varietes, developed by Gillet and Soulé and generalized to the case of line bundles with logarithmically singular metrics by Burgos, Kramer, and Kühn, states a formula for the arithmetic self intersection number ϖ4 in terms of integrals of Green currents over certain cycles on the complex fibre of A2, and a contribution from the finite fibres. The computation of these ingredients for ϖ4 can be tackled by regarding the space A2 from different points of view. We will specify these viewpoints and sketch the associated methods for obtaining the explicit value of ϖ4.
05.06.2018 Anna von Pippich (TU Darmstadt)
Title: An analytic class number type formula for PSL2(Z)
Abstract: For any Fuchsian subgroup Γ ⊂ PSL2(R) of the first kind, Selberg introduced the Selberg zeta function in analogy to the Riemann zeta function using the lengths of simple closed geodesics on Γ\H instead of prime numbers. In this talk, we report on a formula that determines the special value at s = 1 of the derivative of the Selberg zeta function for Γ = PSL2(Z). This formula is obtained as an application of a generalized Riemann--Roch isometry for the trivial sheaf on Γ\H, equipped with the Poincaré metric. This is joint work with Gerard Freixas.
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.
19.06.2018 Alberto Navarro Garmendia (Uni Zürich)
Title: Recent advancements on the Grothendieck-Riemann-Roch theorem
Abstract: Grothendieck's Riemann-Roch theorem compares direct images at the level of K-theory and the Chow ring. After its initial proof at the Borel-Serre report, Grothendieck aimed to generalise the Riemann-Roch theorem at SGA 6 in three directions: allowing general schemes not necessarily over a base field, replacing the smoothness condition on the schemes by a regularity condition on the morphism, and avoiding any projective assumption either on the morphism or on the schemes. After the coming of higher K-theory there was also a fourth direction, to prove the Riemann-Roch also between higher K-theory and higher Chow groups. In this talk we will review recent advancements in these four directions during the last years. If time permits, we will also discuss refinements of the Riemann-Roch formula which takes into account torsion elements.
26.06.2018 kein Seminar
03.07.2018 kein Seminar
10.07.2018 Javier Fresán (École Polytechnique)
Title: Hodge Theory of Kloosterman Connections
Abstract: Broadhurst and Roberts recently studied the L-functions associated with symmetric powers of Kloosterman sums and conjectured a functional equation after extensive numerical computations. By the work of Yun, these L-functions correspond to “usual” motives over Q which, in low degree, are known to be modular. For the purpose of computing the Hodge numbers or relating the L-functions to periods, it is however more convenient to change gears and work with exponential motives. I will construct the relevant motives and show how the irregular Hodge filtration allows one to explain the gamma factors at infinity in the functional equation, as well as to get lower bounds for the p-adic valuations of Frobenius eigenvalues. It is a joint work with Claude Sabbah and Jeng-Daw Yu.
17.07.2018 kein Seminar


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