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

## Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2014/15

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.

 14.10.2014 Vorbesprechung 21.10.2014 kein Seminar 28.10.2014 Niels Lindner (HU Berlin) Bertini theorems for simplicial toric varieties over finite fields Abstract: The classical Bertini theorems on generic smoothness and irreducibility do not hold for varieties over finite fields. However, by the work of Poonen and Charles, it is still possible to give a "probability" for smoothness or geometric irreducibility in certain linear systems on subvarieties of projective space over a finite field. Both versions extend to the context of projective simplicial toric varieties. As an application, one finds that "almost all" hypersurfaces in a nice simplicial toric variety are geometrically irreducible and have a finite singular locus, which is small compared to the degree. 04.11.2014 Barbara Jung (HU Berlin) An integral around the toroidal boundary of A2 Abstract: To obtain the arithmetic degree of the Hodge bundle on a (toroidal) compactification of the Siegel modular variety A2 of degree two, an integral over the regularized star product of Green objects, corresponding to Siegel modular forms, has to be computed. By definition of this product and Stokes’ Theorem, an integral over an ε-tube around the boundary appears. As the integrand degenerates near the boundary, it is not clear that this converges. We will give an explicit description of the geometric situation and show that the integral is in fact going to zero for ε approaching zero. 11.11.2014 kein Seminar 18.11.2014 Benjamin Göbel (Universität Hamburg) Arithmetic local coordinates and applications to arithmetic self-intersection numbers Abstract: In order to calculate the arithmetic self-intersection number of an arithmetic prime divisor on an arithmetic surface, we need to move the prime divisor by the divisor of a rational function. Since there is no canonical choice for the rational function, we may ask whether there is an analytic shadow of the prime divisor that replaces the geometric intersection number at the nite places on the arithmetic surface by an analytic datum on the induced complex manifold. This leads to the denition of an arithmetic local coordinate. In this talk we show that the arithmetic self-intersection number of an arithmetic divisor can be written as a limit formula using an arithmetic local coordinate. We also apply this idea to the intersection theory of H. Gillet and C. Soule and to the generalized intersection theory of J. I. Burgos Gil, J. Kramer and U. Kühn. 25.11.2014 Peter Bruin (Universität Leiden) Optimal bounds for the difference between the Weil height and the Néron-Tate height for elliptic curves over Qbar Abstract: Consider an elliptic curve E over Qbar given by a Weierstraß equation with algebraically integral coefficients. For the purpose of computing Mordell-Weil groups, one would like to bound the difference between the two standard height functions on E(Qbar) (the Weil height and the Néron-Tate height) as sharply as possible. I will describe an algorithm that computes the infimum and the supremum of the difference between these height functions to any desired precision. The main source of difficulties are the Archimedean places; it turns out that these can be treated using the classical Weierstraß elliptic functions. 02.12.2014 Martin Bright (Universität Leiden) The Brauer-Manin obstruction and reduction mod p Abstract: The Brauer-Manin obstruction plays an important role in the study of rational points on varieties. Indeed, for rational varieties (such as cubic surfaces) it is conjectured that this obstruction determines whether or not a variety has a rational point. I will give a brief introduction to the Brauer-Manin obstruction and then look at recent results relating it to the geometry of the variety at primes of bad reduction. 09.12.2014 kein Seminar 16.12.2014 Arne Smeets (KU Leuven/Orsay) On the insufficiency of the étale Brauer-Manin obstruction Abstract: Since Poonen's construction of a variety X defined over a number field k for which X(k) is empty and the \'etale Brauer-Manin set X(\mathbf{A}_k)^\text{Br,\'et} is not, several other examples of smooth, projective varieties have been found for which the \'etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using Poonen's trick'', i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this talk, we construct examples for which the Albanese variety is trivial. The new geometric ingredient in our construction is the appearance of Beauville surfaces. Assuming the abc conjecture and using geometric work of Campana on orbifolds, we also prove the existence of an example which is simply connected. 06.01.2015 kein Seminar 13.01.2015 Maryna Viazovska (HU Berlin) Strongly regular graphs from the geometrical point of view Abstract: A strongly regular graph with parameters (v, k, l, m) is a k-regular graph in which every pair of adjacent vertices has l common neighbors and every pair of non-adjacent vertices has m common neighbors. In this talk we will give an overview of the theory of these graphs. Also we will report on new non-existance results for strongly regular graphs. 20.01.2015 Kay Rülling (FU Berlin) Vanishing of the higher direct images of the structure sheaf Abstract: Let f: X---> Y be a birational and projective morphism between excellent and regular schemes. Then the higher direct images of the structure sheaf of X under f, R^i f_* O_X, vanish for all positive integers i. In case X and Y are smooth schemes over a field of characteristic zero, this vanishing was proved by Hironaka as a corollary of his proof of the existence of resolutions of singularities. In case X and Y are smooth over a field of positive characteristic the statement was proved by Chatzistamatiou-Rülling in 2011. In this talk I will explain the proof in the general case. This is joint work with Andre Chatzistamatiou. 27.01.2015 Rafael von Känel (MPI Bonn) Discriminants and small points of cyclic covers Abstract: Let K be a number field. We first consider a generalization of Szpiro's discriminant conjecture to arbitrary smooth, projective and geometrically connected curves X/K of positive genus. Then we present an unconditional exponential version of this conjecture for cyclic covers of the projective line, and we discuss a related work (jointly with A. Javanpeykar) in which we established Szpiro's small points conjecture for cyclic covers. We also plan to explain the proofs. They combine the theory of logarithmic forms with Arakelov theory for arithmetic surfaces. 03.02.2015 ACHTUNG andere Uhrzeit: 13:00 - 14:00 Uhr Henrik Bachmann (Universität Hamburg) Multiple zeta values and multiple Eisenstein series Abstract: In the first half of the talk I will introduce multiple zeta values and discuss their algebraic structure. Multiple zeta values can be seen as a multiple version of the Riemann zeta values appearing in different areas of mathematics and theoretical physics. The product of these real numbers can be expressed in two different ways, the so called stuffle and shuffle product, which yields a large family of linear relations. The second part of the talk is dedicated to multiple Eisenstein series which can be seen as a multiple version of the classical Eisenstein series for the full modular group. By definition the multiple Eisenstein series functions also fulfill the stuffle product. I will explain a recent result which solves the problem of getting also the shuffle product for these functions. 10.02.2015 kein Seminar

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Wintersemester 2007/08
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