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

Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2013/14

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.

22.10.2013 Jürg Kramer (HU Berlin)
Effective bounds for Faltings's delta function
Abstract: In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces. For a given compact Riemann surface X of genus g, this invariant is roughly given as minus the logarithm of the distance of the point in the moduli space of genus g curves determined by X to its boundary. In our talk, we will first give a formula for Faltings's delta function for compact Riemann surfaces of genus g>1 in purely hyperbolic terms. This formula will then enable us to deduce effective bounds for Faltings's delta function in terms of the smallest non-zero eigenvalue and the shortest closed geodesic of X. If time permits, we will also address a question of Parshin related to bounding the height of rational points on curves defined over number fields.
29.10.2013 Ana Maria Botero (HU Berlin)
Spherical varieties
Abstract: First, we will introduce the notion of spherical varieties and discuss important subclasses (horospherical, toroidal) and many examples of them. We present their description by so-called colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as T-varieties. Thus, colored fans turn into p-divisors. The latter is recent work by Klaus Altmann, Valentina Kiritchenko and Lars Petersen.
05.11.2013 kein Seminar
12.11.2013 Giovanni De Gaetano (HU Berlin)
Towards an arithmetic Riemann Roch theorem for non-compact modular curves
Abstract: The goal of the talk is to survey the construction and the explicit computation of the Quillen metric on the determinant of cohomology of powers of the Hodge bundle on a modular curve, this corresponds to the definition and the computation of the left hand side of an analogue of Gillet and Soulè's arithmetic Riemann Roch theorem for smooth arithmetic surfaces. In specific we want to describe how the case of the first power of the Hodge bundle is problematic in this situation. If time permits we would like to discuss the right hand side of the formula in the already solved cases, and specify what our task in this context would be.
19.11.2013 Remke Nanne Kloosterman (HU Berlin)
Alexander polynomials of curves and Mordell-Weil ranks of Abelian Varieties
Abstract: Let $C=\{f(z_0,z_1,z_2)=0\}$ be a plane curve with ADE singularities. Let $m$ be a divisor of the degree of $f$ and let $H$ be the hyperelliptic curve \[ y^2=x^m+f(s,t,1).\] defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of $H$. For this we use the Alexander polynomial of $C$. This extends a result by Cogolludo-Augustin and Libgober for the case of where $C$ is a curve with ordinary cusps. In the second part of the talk we sketch how the work of David Ouwehand can be used to obtain a similar result for Jacobians of curves over $\mathbb{F}_q(s,t)$.
26.11.2013 José Burgos (ICMAT, Madrid)
Equidistribution of small points on toric varieties
Abstract: As the culmination of work of many mathematicians, Yuan has obtained a very general equidistribution result for small points on arithmetic varieties. Roughly speaking Yuan's theorem states that given a "very" small generic sequence of points with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to the measure associated to the hermitian line bundle. Here very small means that the height of the points converges to the lower bound of the essential minimum given by Zhang's inequalities. The existence of a very small generic sequence is a strong condition on the arithmetic variety because it implies that the essential minimum attains its lower bound. We will say that a sequence is small if the height of the points converges to the essential minimum. By definition every arithmetic variety contains small generic sequences. We show that for toric line bundles on toric arithmetic varieties Yuan's theorem can be split in two parts: a) Given a small generic sequence of points, with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to a measure. b) If the sequence is very small, the limit measure agrees with the measure associated to the hermitian line bundle.
03.12.2013 kein Seminar
10.12.2013 Aprameyo Pal (Uni Heidelberg, zzt. HU Berlin)
Non-commutative Iwasawa theory and its applications
Abstract: In Arithmetic Geometry one of the main themes has always been to understand the interplay between analytic invariants and algebraic invariants. One of the most famous example of this interplay is Birch and Swinnerton-Dyer conjecture. Iwasawa theory is one of the important tools which sheds some light on this issue. It provides a crucial link between the characteristic ideal of the Selmer groups (which are defined algebraically) and p-adic L-functions (which are defined analytically). In this talk, I will explain how non-commutative Iwasawa theory fits in the bigger picture. I will explain some results in function field and number field case. If time permits, I will sketch some proofs.
17.12.2013 Wojciech Gajda (Adam Mickiewicz University, Poznań)
Independence of \ell-adic representations
Abstract: We discuss certain arithmetical properties of Galois representations attached to etale cohomology of algebraic varieties and schemes defined over finitely generated fields of any characteristic. The talk will contain a report on recent joint work with Gebhard Boeckle and Sebastian Petersen.
07.01.2014 kein Seminar
14.01.2014 Frank Gounelas (HU Berlin)
Cohomology of SRC varieties in positive characteristic
Abstract: This talk will offer two different perspectives on the fact that a separably rationally connected variety in characteristic p has H^1(X, O_X)=0. Over C, this result follows from Hodge theory. The first proof comes from a result of Biswas-dos Santos on triviality of vector bundles on SRC varieties (following from deep recent results of Langer in char p), whereas the second is cohomological (etale and crystalline) in nature. I will introduce some of the necessary background theory and try to include full proofs.
21.01.2014 kein Seminar
28.01.2014 James Stankewicz (University of Copenhagen)
Palindromic Properties and Descent Obstructions
Abstract: For most curves that you might think of, it is possible to find a twist which has a rational point. For the first time we exhibit an infinite family of curves over the rational numbers for which this explicitly does not apply. That is to say that we find Shimura curves C whose lack of rational points is palindromic or preserved by twists. Using this family of curves, we find a related set of twists of Shimura curves which all violate the Hasse Principle. This violation is explicitly given by a descent obstruction.
04.02.2014 Tim Dokchitser (University of Bristol)
L-functions of curves
Abstract: L-functions of elliptic curves have been studied a lot and their local invariants (local factors, conductors, Tate module, root numbers etc.) are well-understood, both theoretically and computationally. For curves of higher genus the situation is more complicated, and I will report on a joint work in progress with Vladimir Dokchitser that attempts to develop the corresponding theory and classification. Basically, there are two approaches to understand these L-functions, one using regular models and one using semistable models. I will explain what they are and what they can achieve, focussing in particular on hyperelliptic curves over the rationals.
11.02.2014 Anna von Pippich (TU Darmstadt)
An arithmetic Riemann-Roch theorem for weighted pointed curves
Abstract: In this talk, we report on work in progress with G. Freixas generalizing the arithmetic Riemann-Roch theorem for pointed stable curves to the case where the metric is allowed to have conical singularities at the marked points. We will first outline the main ideas of the proof and then focus on some analytical ingredients, e.g. the explicit computation of the regularized determinant for hyperbolic cusps and cones.

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