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

## Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2015

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.

 14.04.2015 Vorbesprechung 21.04.2015 kein Seminar 28.04.2015 Vortrag von Bas Edixhoven (Universität Leiden) im Gästeseminar "Arithmetische Geometrie" der FU Berlin 05.05.2015 kein Seminar 12.05.2015 Yingkun Li (TU Darmstadt) Special values of Green function at twisted big CM points Abstract: A Green function on an arithmetic variety is a function with logarithmic singularity along an algebraic divisor. Their values at CM points play an important role in the theory of arithmetic intersection. In the case of Hilbert modular surface, one could use Poincare series to explicitly construct the Green function with log singularity along Hirzebruch-Zagier divisors. Its values averaging over Galois orbits of a big CM point are rational numbers and have interesting factorizations. In this talk, we will recall these notions and use harmonic Maass forms of weight one to give a modular interpretation of the values of these Green function at twisted big CM points. 19.05.2015 kein Seminar 26.05.2015 Wouter Zomervrucht (Universität Leiden) Bhargava's cube law and cohomology Abstract: In his Disquisitiones Arithmeticae, Gauss described a composition law on (equivalence classes of) integral binary quadratic forms of fixed discriminant D. The resulting group is the class group Cl(S), where S is the quadratic algebra of discriminant D. More recently, Bhargava explained Gauss composition as a consequence of a composition law on (equivalence classes of) 2x2x2-cubes of integers. Here one obtains the group Cl(S) x Cl(S). Bhargava's proof is arithmetic. We show how to obtain Bhargava's cube law instead from geometry, with the class groups arising as cohomology. This is work in progress. 02.06.2015 Andriy Bondarenko (Norwegian University of Science and Technology) On Helson's conjecture Abstract: [see here] 09.06.2015 Jürg Kramer (HU Berlin) Degeneration of the hyperbolic heat kernel Abstract: In our talk we will investigate the degeneration of the hyperbolic heat kernel and its trace at the cusps of modular curves. This degeneration behavior should be similar to the degeneration of the arithmetic self-intersection number of the corresponding line bundle equipped with a metric that is logarithmically singular at the cusps. 16.06.2015 Valentina Di Proietto (FU Berlin) On the homotopy exact sequence for the log algebraic fundamental group Abstract: There is a strong link between the fundamental group of a variety and the linear differential equations we can define on it. The definition of the fundamental group given in terms of homotopy classes of loops does not generalize easily to algebraic varieties defined over an arbitrary field. But exploiting this link we can give an other definition that makes sense in very general contexts: it is called the algebraic fundamental group. We prove the homotopy exact sequence for the algebraic fundamental group for a fibration with singularities with normal crossing and we explain how this gives a monodromy action. This is a joint work with Atsushi Shiho. 23.06.2015 Slawomir Rams (Jagiellonian University Krakow, z.Zt. Leibniz Universität Hannover) On Enriques surfaces with four cusps Abstract: One can show that maximal number of A2-configurations on an Enriques surface is four. In my talk I will classify all Enriques surfaces with four A2 -configurations. In particular I will show that they form two families in the moduli of Enriques surfaces and I will construct open Enriques surfaces with fundamental groups (Z/3Z)^2 × Z/2Z and Z/6Z, completing the picture of the A2 -case and answering a question put by Keum and Zhang. This is joint work with M. Schuett/LUH Hannover. 30.06.2015 Antareep Mandal (HU Berlin) Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms Abstract: The average over an orthonormal basis of cusp forms on a given Siegel modular curve can be viewed as the lower part of the discrete spectrum, corresponding to the eigenvalue 0, of the heat kernel associated to the Siegel-Maaß Laplacian. Therefore, one can attempt to use long time asymptotics of this heat kernel to derive an optimum sup-norm bound for the average over an orthonormal basis of cusp forms uniformly in a finite degree cover of the given Siegel modular curve. This method has been proven to work in the one-dimensional case of classical modular forms. In this talk, we present our progress towards a generalization of this method to the higher dimensional case of Siegel modular forms. 14.07.2015 Ariyan Javanpeykar (Uni Mainz) Good reduction of complete intersections Abstract: In 1983, Faltings proved the Shafarevich conjecture: for a finite set of finite places of a number field K and an integer g>1, the set of isomorphism classes of curves of genus g over K with good reduction outside S is finite. In this talk we shall consider analogues of the Shafarevich conjecture for complete intersections. This is joint work with Daniel Loughran.

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Wintersemester 2007/08
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