Sommersemester 2012

 10.04.2012 Vorbesprechung 17.04.2012 Jakob Scholbach (Univ. Münster) Motives and Arakelov theory Abstract: Beginnend mit einer kurzen Einführung in die stabile Homotopiekategorie von Schemata werden wir eine neue Kohomologietheorie namens Arakelov-motivischer Kohomologie vorstellen. Diese kann als Variante (und Verallgemeinerung) von arithmetischen K- und Chow-Gruppen angesehen werden. Wir diskutieren einige Eigenschaften wie den arithmetischen Satz von Riemann-Roch sowie, falls Zeit bleibt, die Beziehung zu speziellen L-Werten. 24.04.2012 Emmanuel Ullmo (Université Paris-Sud, Orsay) The hyperbolic Ax-Lindemann conjecture for projective Shimura varieties and some applications to the André-Oort conjecture Beginn: 14.15 Uhr 08.05.2012 Nuno Freitas (Univ. Barcelona) The generalized Fermat-type equations x5+y5 =2zp or 3zp via Q-curves Abstract: In order to attack the generalized Fermat equation Axp+Byq =Czr the modular approach to Diophantine equations that initially led to the proof of Fermat's Last Theorem has been progressively refined. In this talk we will explain how several pieces of the strategy need to be generalized in order to solve equations of the form x5+y5 =Czp. In particular, we will show how we can use two simultaneous Frey-curves defined over Q(√5) to solve the previous equations for a set o primes with density 3/4. 15.05.2012 Anna von Pippich (HU Berlin) On the spectral zeta function of a hyperbolic cusp or cone Abstract: In a joint project with G. Freixas we aim at establishing an arithmetic Riemann-Roch isometry for singular metrics. In this talk we report on an analytic ingredient, namely the computation of the regularized determinant of the hyperbolic Laplacian on a cusp or cone with Dirichlet boundary conditions. 22.05.2012 kein Seminar 29.05.2012 Christian Liedtke (Univ. Bonn) On the Birational Nature of Lifting Abstract: Whenever a variety X lifts from characteristic p to characteristic zero, say over the Witt ring, then many classical results over the complex numbers hold for X, and certain "characteristic p pathologies" cannot occur, simply because one can reduce modulo p (I will discuss this in examples). But then, lifting results are difficult, and generally, varieties do not lift. However, in many situations, it is possible or easier to lift a birational model of X, maybe even one that has "mild" singularities (again, I will give examples). So, a natural question is whether the liftability of such a birational model implies that of our original X. We will show that this completely fails in dimension at least 3, that this question is surprisingly subtle in dimension 2, and that it is trivial in dimension 1. 05.06.2012 Andrea Cattaneo (Univ. Mailand) Elliptic CY threefolds over surfaces Abstract: In this talk I want to give a classification of the elliptic CY threefolds we can find in a projective bundle over a base surface B. More in detail, if L is an ample line bundle on B, then I want to give explicit bounds on the pairs (a,b) such that in the bundle P(O + La +Lb) the generic anticanonical variety is smooth. I will also give a detailed description in the case B = P2. As an application of this classification, I will switch to physics and show a nice result concerning string theory, which generalizes a result by Aluffi-Esole. 12.06.2012 Wouter Castryck (Univ. Leuven) Curve gonalities and Newton polygons Abstract: Let Delta be a lattice polygon, i.e. the convex hull in R2 of a finite number of points of Z2 (called "lattice points"). Assume that it is not contained in a line. Then it is well-known that a generic Laurent polynomial f(x,y) having Delta as its Newton polygon defines a curve whose genus equals the number of lattice points in the interior of Delta. In this talk we will search for combinatorial interpretations for other discrete invariants, such as the Clifford index, Clifford dimension, and the gonality. The latter is by definition the minimal degree of a morphism to P1. 19.06.2012 Hannah Enders (HU Berlin) On perfect sheaves of modules Abstract: In 1957, Rees defined perfect modules as modules where the homological invariants grade and projective dimension coincide. Starting from this point, we will investigate under which conditions a definition of perfect sheaves of modules is possible. Furthermore, we will show that perfect sheaves of modules inherit some interesting properties of perfect modules. 26.06.2012 Jürg Kramer (HU Berlin) Remark on a question of Parshin 03.07.2012 Barbara Jung (HU Berlin) Computations towards the arithmetic self intersection number of the bundle of modular forms on A2 Abstract: To compute the self intersection number of the bundle of modular forms on A2, one has to choose sections tactically, as one has to integrate over forms and subvarieties defined by them. I will present a computation of one of the arising terms via induction A1 → A2. 10.07.2012 Lars Kühne (ETH Zürich) An effective result of André-Oort type Abstract: "The André-Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety means an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue of the classical Manin-Mumford conjecture on the distribution of torsion points in abelian varieties. I will present a rarely known approach to the AOC that goes back to Yves André himself: Before the model-theoretic proofs of the AOC in certain cases by the Pila-Wilkie-Zannier approach, André presented the first non-trivial proof of the AOC in case of a product of two modular curves. In my talk, I discuss results in the style of André's method, allowing to actually compute all special points in a non-special curve of a product of two modular curves and more..."