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

## Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2016/17

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.

 18.10.2016 Vorbesprechung 25.10.2016 kein Seminar 01.11.2016 Antareep Mandal (HU Berlin) Title: Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms Abstract: This talk updates our progress towards obtaining the optimal sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. In particular, we study Maass operators and generalize the relation between the cusp forms and the Maass forms in higher dimensions along with presenting an interesting trick to obtain the heat kernel corresponding to the Maass Laplacians by adapting the "method of images" used to obtain the heat kernel corresponding to the Laplace-Beltrami operator. 08.11.2016 Erik Visse (University of Leiden) Title: A heuristic for a Manin-type conjecture for K3 surfaces Abstract: In the late 1980's Manin came up with a conjecture for the growth of rational points of bounded height of Fano varieties; he predicted what the number of rational points on a suitable open subset of any given such variety should be asymptotically. Since Manin's original paper, many specific cases have been studied giving rise to refinements, proofs, upper and lower bounds, counterexamples and proposed fixes; all still concerning Fano varieties. In my PhD project, I study the same problem for the "next case" in dimension 2: K3 surfaces. Recently I was able to compute heuristics for certain diagonal quartic surfaces that agree with some numerical experiments that were done by my supervisor Ronald van Luijk a few years ago. In the talk I will explain the techniques involved and some problems that need to be overcome in order to formulate a reasonable conjecture. 15.11.2016 kein Seminar 22.11.2016 Remke Kloosterman (University of Padova) Title: Monomial deformations of Delsarte Hypersurfaces and Arithmetic Mirror Symmetry Abstract: In a recent preprint Doran, Kelly, Salerno, Sperber, Voight and Whitcher study for five distinct quartic Delsarte surfaces a one-parameter monomial deformation. Using character sums they find a remarkable similarity between the zeta functions of general members of each of the families. In this talk we present another approach to prove these results. Moreover, we give a necessary and sufficient combinational criterion to check for a pair of monomial deformations of Delsarte hypersurfaces whether their zeta functions are essentially the same or not. 29.11.2016 Anna von Pippich (TU Darmstadt) Title: A Rohrlich-type formula for the hyperbolic 3-space Abstract: Jensens's formula is a well-known theorem of complex analysis which characterizes, for a given meromorphic function $f$, the value of the integral of $|\log(f(z))|$ along the unit circle in terms of the zeros and poles of $f$ inside this circle. An important theorem of Rohrlich generalizes Jensen's formula for modular functions $f$ with respect to the full modular group, and expresses the integral of $|\log(f(z))|$ over a fundamental domain in terms of special values of Dedekind's Delta function. In this talk, we report on a Rohrlich-type formula for the hyperbolic 3-space. 06.12.2016 Ania Otwinowska (U Paris-Sud, Orsay) Title: Around standard conjectures for algebraic cycles Abstract: Given a (reasonable) topological space X, one can study its shape by defining a simple invariant attached to X: its cohomology. When moreover X is a complex algebraic variety (i.e. the set of zeroes of a finite collection of complex polynomials) one would like to understand the collection of its algebraic cycles, namely the formal linear combinations of its algebraic subvarieties, and their relations with the cohomology of X. In the 60's Grothendieck proposed a set of simple statements describing some naturally expected relations between algebraic cycles and cohomology: the standard conjectures. In characteristic zero they are implied by the Hodge conjecture. Voisin proved that the following conjecture is a consequence of the standard conjectures: Conjecture N (Voisin): Let X be an algebraic variety. If Z is an algebraic cycle in X whose cohomology class is supported on a closed subvariety Y, then Z is homologically equivalent to a cycle supported on Y. In the first part of this talk, I will present in an elementary way the notion of algebraic cycle, and the above conjectures. In the second part, I will explain a converse to Voisin's result: Theorem (O.) In characteristic 0, Voisin's conjecture N is equivalent to the standard conjectures. 17.01.2017 Das Seminar findet am 17.01. ausnahmsweise von 15:00-16:30 in Raum 1.023, RUD 25 statt. Yiannis Petridis (University College London) Title: Lattice point problems in hyperbolic spaces Abstract: [see here] 24.01.2017 Sebastián Herrero (Chalmers Tekniska Högskola/University of Gothenburg) Title: Asymptotic distribution of Hecke points over Cp Abstract: [see here] 07.02.2017 Bas Edixhoven (University of Leiden) Title: Group schemes out of birational group laws Abstract: In his construction of the jacobian variety of a smooth projective algebraic curve C over a field, Weil first showed that the gth symmetric power of C (with g the genus of C) has a birational group law, and then that this birational group law extends uniquely to a group variety. In the 1960's, Weil's extension result was generalised to schemes by Michael Artin in SGA 3, and used for the construction of reductive group schemes and of Neron models of abelian varieties. At the occasion of the re-edition of SGA 3, I had a look at Artin's article, and it seemed to me that it was better to change the approach to the problem. The main improvement is to construct the group scheme as a sub sheaf of an fppf sheaf of relative birational maps. The fact that relative birational maps admit fppf descent seems to be new. As a consequence, some finiteness conditions in Artin's article are no longer needed. Joint work with Matthieu Romagny: http://arxiv.org/abs/1204.1799 Appeared in Panor. Synthèses, 47, Soc. Math. France, Paris, 2015. 14.02.2017 Joachim Mahnkopf (Universität Wien) Title: On local constancy of dimension of slope subspaces of automorphic forms Abstract: We prove a higher rank analogoue of a Conjecture of Gouvea-Mazur on local constancy of dimension of slope subspaces of automorphic forms for reductive groups having discrete series. The proof is based on a comparison of Bewersdorff's elementary trace formula for pairs of congruent weights and does not make use of p-adic Banach space methods or rigid analytic geometry.

Archiv:

Wintersemester 2007/08
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