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

Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2016

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.

19.04.2016 Vorbesprechung
26.04.2016 kein Seminar
03.05.2016 kein Seminar
10.05.2016 Maryna Viazovska (HU Berlin)
Title: Modular forms and sphere packing
Abstract: In this talk we will repot on our recent result on the sphere packing problem in dimensions 8 and 24. We will explain the linear programming method for sphere packing intoduced by N. Elkies and H. Cohn. Also we will present the constuction of certificate functions providing the optimal estimate for the sphere packing problem in dimensions 8 and 24.
17.05.2016 Anton Mellit (SISSA Trieste)
Title: Mixed Hodge structures of character varieties.
Abstract: The conjecture of Hausel, Letellier and Villegas gives precise predictions for mixed Hodge polynomials of character varieties. In certain specializations this conjecture also computes Hurwitz numbers, Kac's polynomials of quiver varieties, and zeta functions of moduli spaces of Higgs bundles. I will formulate the conjecture, give some examples, and talk about my proof of polynomiality of the generating functions that arise there.
24.05.2016 Barbara Jung (HU Berlin)
Title: The arithmetic volume of the moduli stack A2
Abstract: The arithmetic volume of the (compactified) moduli stack An/Z of principally polarized n-dimensional abelian varieties is given by the arithmetic self intersection number of the bundle of Siegel modular forms on An, metrized by the Petersson norm. A generalized intersection theory applicable for this case was developed by Burgos, Kramer and Kühn in 2005. It is conjectured that the above intersection number consists of a sum of special values of the logarithmic derivative of the zeta function. We will present a way to compute the volume of A2, using results from Kudla and Kühn, and discuss how to handle the boundary.
31.05.2016 Fabien Pazuki (University of Copenhagen)
Title: Bad reduction of curves with CM jacobians
Abstract: An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in the entry $[I_0-I_0-m]$ in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves.

We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over $\overline{\mathbb{Q}}$ with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit an infinite family of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of $\mathbb{Q}$ of degree 4 that contains $\mathbb{Q}(\sqrt{5})$
07.06.2016 kein Seminar
Birthday conference "Arakelov theory and automorphic forms"
14.06.2016 Jürg Kramer (HU Berlin)
Title: On the volume of the Siegel modular variety
Abstract: In our talk we provide a short proof of Siegel's formula for the volume of the Siegel modular variety. The proof makes essential use of the Klingen Eisenstein series and is based on a geometric induction argument.
21.06.2016 Fabian Völz (TU Darmstadt)
Title: Elliptic and hyperbolic Eisenstein series as theta lifts
Abstract: Generalising the concept of classical non-holomorphic Eisenstein series associated to cusps, one can define elliptic Eisenstein series associated to points in the upper-half plane, and hyperbolic Eisenstein series associated to geodesics. In my talk I will show that averaged versions of these elliptic and hyperbolic Eisenstein series can be obtained as theta lifts of signature (2,1) of some weighted Poincaré series, which generalizes a classical result in the parabolic case. Moreover, I will show how to realize a distinguished elliptic Eisenstein series as a theta lift of signature (2,2). Finally, if time permits, I will propose some applications of these results.
28.06.2016 kein Seminar
05.07.2016 Ana Maria Botero (HU Berlin)
Title: Intersection theory of b-divisors on toric varieties
Abstract: We introduce toric b-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions, toric b-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b-divisor corresponds to the number of lattice points in this convex set and we give a Hilbert-Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. We further investigate the question of extending these results to arbitrary toroidal varieties. Examples in which such b-divisors naturally appear are invariant metrics on line bundles over toroidal compactifications of mixed Shimura varieties. Indeed, the singularity type which the metric acquires along the boundary can be encoded using toroidal b-divisors.
12.07.2016 Martin Westerholt-Raum (Chalmers University of Technology)
Bitte beachten: Das Seminar findet heute ausnahmsweise um 15:00-16:30 in Raum 1.023 (BMS Seminar Room) statt.
Title: Automorphic constituents of tensor products of Harish-Chandra modules
Abstract: Products of real-analytic automorphic forms in general generate more than one automorphic representation. We study this phenomenon at the infinite place for scalar valued Siegel modular forms of genus 2. It turns out that automorphic constituents of the specific tensor products that we inspect are all holomorphic (limits) of discrete series. This has applications to Poincaré series and harmonic weak Siegel Maaß forms.


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