Humboldt Universität zu Berlin
Mathem.Naturwissenschaftliche Fakultät
Institut für Mathematik
Sommersemester 2018
Das Forschungsseminar findet dienstags in der Zeit von 13:15  15:00 Uhr in der Rudower Chaussee 25, 12489 BerlinAdlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.
24.04.2018  Ulf Kühn (Uni Hamburg) 
Title: Multiple qzeta values and period polynomials Abstract: We present a class of qanalogues of multiple zeta values given by certain formal qseries with rational coefficients. After introducing a notion of weight and depth for these qanalogues of multiple zeta values we will state a dimension conjecture for the spaces of their weight and depthgraded 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 NéronTate 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 socalled NéronTate height. The NéronTate height is always nonnegative, and behaves well with respect to multiplicationbyN on A. Its good properties have led and still lead to applications in number theory. We discuss formulas for the NéronTate 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 R_{g} of Picard numbers of gdimensional 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 A_{2} Abstract: The Hodge line bundle ω on the moduli stack A_{2}, metrized by the L^{2}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 ϖ^{4} in terms of integrals of Green currents over certain cycles on the complex fibre of A_{2}, and a contribution from the finite fibres. The computation of these ingredients for ϖ^{4} can be tackled by regarding the space A_{2} from different points of view. We will specify these viewpoints and sketch the associated methods for obtaining the explicit value of ϖ^{4}. 

05.06.2018  Anna von Pippich (TU Darmstadt) 
Title: An analytic class number type formula for PSL_{2}(Z) Abstract: For any Fuchsian subgroup Γ ⊂ PSL_{2}(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 = PSL_{2}(Z). This formula is obtained as an application of a generalized RiemannRoch 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 Qvector 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,d1), 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 padic places. It turns out that this question is more tractable. Moreover, using a classical product formula on quadratic forms, the padic result will give us nontrivial 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 GrothendieckRiemannRoch theorem Abstract: Grothendieck's RiemannRoch theorem compares direct images at the level of Ktheory and the Chow ring. After its initial proof at the BorelSerre report, Grothendieck aimed to generalise the RiemannRoch 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 Ktheory there was also a fourth direction, to prove the RiemannRoch also between higher Ktheory 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 RiemannRoch 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 Lfunctions associated with symmetric powers of Kloosterman sums and conjectured a functional equation after extensive numerical computations. By the work of Yun, these Lfunctions 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 Lfunctions 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 padic valuations of Frobenius eigenvalues. It is a joint work with Claude Sabbah and JengDaw Yu. 

17.07.2018  kein Seminar 
Wintersemester 2007/08
