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

Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2015/16

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.

13.10.2015 Vorbesprechung
20.10.2015 kein Seminar
27.10.2015 Davide Veniani (Leibniz Universität Hannover)
Counting lines on quartic surfaces: New techniques and results
Abstract: In the last years, two independent research teams (the one based in Ankara, Turkey, the other in Hannover, Germany) have tackled the problem of counting lines on smooth quartic surfaces; the former aimed at a complete classification using computer algebra system GAP, the latter strove for more geometrical insight. The synergy between the two methods has fostered new ideas towards three goals: (1) finding a proof of the fact that the maximal number of lines is 64 which does not involve the flecnodal divisor; (2) proving the uniqueness of the surface with 64 lines with a geometrical approach; (3) adapting the methods to the K3 quartic case. I will report about the state of the art.
03.11.2015 Anilatmaja Aryasomayajula (University of Hyderabad)
Title: Heat kernels, Bergman kernels, and estimates of cusp forms
Abstract: In this talk, we describe a geometric approach to study estimates of cusp forms. The approach relies on the micro-local analysis of the heat kernel and the Bergman kernel. Using which we can derive qualitative estimates of cusp forms of integral weight or half-integral weight associated to arbitrary Fuchsian subgroups and groups commensurable with the Hilbert modular group.
10.11.2015 kein Seminar
17.11.2015 Anna von Pippich (TU Darmstadt)
Title: On the wave representation of Eisenstein series
Abstract: Let Γ ⊂ PSL2 (ℝ) be a Fuchsian subgroup of the first kind and let X = Γ\H be the associated finite volume hyperbolic Riemann surface. Eisenstein series attached to parabolic subgroups of Γ play a fundamental role in the theory of automorphic forms on X. Analoguously, one can consider Eisenstein series associated to hyperbolic or elliptic subgroups of Γ. In this talk, we present a unified approach to the construction of these Eisenstein series in terms of the wave kernel. This is joint work with Jay Jorgenson and Lejla Smajlovi'.
24.11.2015 David Holmes (University of Leiden)
Title: Néron models over bases of higher dimension
Abstract: Néron models for 1-parameter families of abelian varieties were defined and constructed by Néron in the 1960’s, and provide a `best possible’ model for the degenerating family. For a degenerating family of abelian varieties over a base scheme of dimension greater than 1, it is much less clear what the `best possible' model for the family would be. If one naively extends Néron’s original definition to this setting then these objects fail to exist, even if we allow blowups or alterations of the base space of the family - more precisely, we give a combinatorial characterisation of exactly when such Néron models of jacobians exist. In the case of the jacobian of the universal curve we will describe the minimal base-change required in order that a Néron model exist, giving a possible answer to the shape of the `best possible degeneration’.
01.12.2015 Daniel Loughran (Leibniz Universität Hannover)
Title: The Hasse norm principle for abelian extensions
Abstract: A classical theorem of Hasse states that, for a cyclic extension of number fields L/K, an element of K is a norm from L if and only if it becomes a norm over all completions of K. In this talk, we study the extent to which this "Hasse norm principle" holds for other abelian extensions. Namely, the distribution of abelian extensions of bounded discriminant that fail the Hasse norm principle. This is joint work with Christopher Frei and Rachel Newton.
08.12.2015 Maryna Viazovska (HU Berlin)
Title: An application of the theory of automorphic forms to discrete geometry
Abstract: In this talk we will give an overview of classical and recent results on energy optimization problems in discrete geometry. We will focus on the interplay of the theory of automorphic forms and Fourier analysis and their applications to discrete geometry.
15.12.2015 kein Seminar
05.01.2015 Danylo Radchenko (MPIM Bonn)
Title: Higher cross-ratios and functional equations for polylogarithms
Abstract: A well-known conjecture of Zagier states that the value of a Dedekind zeta function of a number field at an integer m>1 can be expressed in terms of the m-th polylogarithm function. This conjecture remains widely open for all m>3. A general strategy for proving it was outlined by Goncharov, and the main ingredient involves constructing certain higher-dimensional generalizations of the classical cross-ratio. In this talk I will give a general definition of higher cross-ratios, show how they can be used to construct interesting functional equations for polylogarithms, and report on the recent progress towards proving Zagier's conjecture in case m=4.
12.01.2015 Jürg Kramer
Title: On the analytic continuation of the heat kernel
Abstract: In our talk we will present an approach of how to analytically continue the heat kernel associated to the Laplacian of quotient spaces of the hyperbolic plane associated to Fuchsian subgroups of the first kind of PSL(2,R).
19.01.2015 Jan Hendrik Bruinier (TU Darmstadt)
Title: Classes of Heegner divisors in generalized Jacobians
Abstract: In parallel to the Gross-Kohnen-Zagier theorem, Zagier proved that the traces of the values of the j-function at CM points are the coefficients of a weakly holomorphic modular form of weight 3/2. Later this result was generalized in different directions and also put in the context of the theta correspondence. We recall these results and report on some newer aspects, which arise from considering classes of Heegner divisors in generalized Jacobians. This is joint work with Y. Li.
26.01.2015 Amir Džambić (Kiel)
Title: The cohomology of the smallest Hurwitz ball quotient
Abstract: Recently, M. Stover showed that there exists the unique compact arithmetic 2-dimensional ball quotient of smallest volume. Its smooth Galois coverings, called Hurwitz ball quotients, thus have the maximal automorphism group among the arithmetic ball quotients with the same Euler number. We study the smallest Hurwitz ball quotient and use the knowledge of the automorphisms and the fundamental group to determine its Picard number and the Albanese variety and study some other of its cohomological properties (joint work with Xavier Roulleau).
02.02.2015 Cecília Salgado (Universidade Federal do Rio de Janeiro, z.Zt. Leibniz Universität Hannover)
Title: Classification of elliptic fibrations on certain K3 surfaces
Abstract: Let X be an algebraic K3 surface endowed with a non-symplectic involution. We classify all elliptic fibrations on X under some hypothesis on the non-symplectic involution. The idea behind it involves transferring the classification problem to a ``simpler'' surface from the geometric point of view. This is work in progress with Alice Garbagnati (Milano).
09.02.2015 kein Seminar


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