Wintersemester 2011/12

 18.10.2011 Vorbesprechung 25.10.2011 Remke N. Kloosterman (HU Berlin) Zariski triples and equisingular deformations of cuspidal curves Abstract: In this talk we construct a Zariski triple, i.e., three plane curves $C_1,C_2,C_3$ of the same degree, with the same number and type of singularities such that the fundamental groups of $\mathbb{P}^2\setminus C_i$ are pairwise non-isomorphic. This is done by calculating the Alexander polynomial of $C_i$. We use this example of a Zariski triple to construct families of singular plane curves such that their equisingular deformation space has larger dimension than expected. In the end we show that if $C$ is a curve of degree at least 13 with non-constant Alexander polynomial then the tangent space of the equisingular deformation space has larger dimension than expected. 01.11.2011 Giovanni De Gaetano (HU Berlin) On the fundamental group of the affine line in positive characteristic Abstract: In the first part of the talk we introduce the notion of fundamental group of a scheme by a pure categorical point of view. This construction is a generalization of the fundamental group of a topological space and of the Galois group of a field. Then, using a little étale topology, we move the notions of "loop" and "neighborhood" from the topological to the arithmetic context. In the last part we see what kind of coverings of the affine line arise when we allow a certain ramification index to be divisible by the characteristic of the field, and we outline a strategy for the solution of the problem. Eventually we will understand that the affine line, in positive characteristic, is very far from being simply connected. 08.11.2011 Ulrich Terstiege (Univ. Duisburg-Essen) Intersections of special cycles on unitary Rapoport-Zink spaces of signature (1,n-1) Abstract: We discuss results on intersections of special cycles on unitary Rapoport-Zink spaces that can be applied to the conjectures of Kudla and Rapoport on intersections of special cycles on unitary Shimura varieties and to the arithmetic fundamental lemma conjecture of W. Zhang. 15.11.2011 Philipp Habegger (Univ. Frankfurt am Main) Beyond the André-Oort Conjecture (joint with Jonathan Pila) Abstract: An isomorphism class of elliptic curves defined over C can be identified with a complex number by virtue of Klein's j-function. The so-called singular j-invariants are particularly interesting from an arithmetic point of view. These come from elliptic curves with complex multiplication. A particular case of the André-Oort Conjecture describes the distribution of points on subvarieties of affine n-space whose coordinates are singular j-invariants. Here the conjecture is known due to work of André, Edixhoven, and Pila. Pink's more general conjecture describes points on subvarieties that satisfy moduli theoretic properties which are generally weaker than asking for complex multiplication. This includes examples such as points in affine n-space whose coordinates are pairwise isogenous elliptic curves. I will present progress into the direction of Pink's Conjecture. Our method of proof relies on the theory of o-minimal structures which has its origins in model theory; the general strategy was developed originally by Zannier. In the talk I will explain what an o-minimal structure is and how it interacts with arithmetic components of our proof. 22.11.2011 Jürg Kramer (HU Berlin) A note on scattering matrices 29.11.2011 Stefan Keil (HU Berlin) Shafarevich-Tate groups of non-square order Abstract: The order of the Shafarevich-Tate group (=sha) of an elliptic curve over a number field is, if it is finite, a square number. For abelian varieties in higher dimensions this is no longer the case. However, for principally polarized abelian varieties over a number field this is almost true, since then the order of sha is a square or twice a square. The only known example of an abelian surface over QQ having order of sha not equal to a square or twice a square has order of sha equal to three times a square. We will explain how to construct abelian surfaces over QQ having order of sha equal to five or seven times a square. 06.12.2011 Matthias Schütt (Univ. Hannover) Modularity of Maschke's octic and Calabi-Yau threefold Abstract: Maschke's octic surface is the unique invariant of a particular group of size 11520 acting on projective fourspace. Recently Bini and van Geemen studied this surface and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group. In particular they computed a decomposition of the cohomology in terms of the group and conjectured its modularity. We will sketch how to actually prove this for all three varieties. The proofs rely on automorphisms of the varieties and in one case on isogenies of K3 surfaces. 13.12.2011 Jeng-Daw Yu (Taiwan National Univ. z.Zt. Essen) The irregular Hodge filtration Abstract: This is a report of work in progress with Esnault. Motivated by various cohomology theories of exponential sums over finite fields, we propose a Hodge-type filtration on the cohomology attached to an exponentially twisted de Rham complex over a complex smooth (quasi-projective) variety. I will indicated some good properties of this filtration and list some natural questions. 03.01.2012 Gérard Freixas i Montplet (Univ. Paris 6) On Faltings theorem for abelian schemes over arithmetic surfaces Abstract: In this talk I will review the statement of the Tate conjecture for abelian varieties. In the number field case this is the celebrated theorem of Faltings. Faltings himself showed how to reduce the case of abelian schemes over higher dimensional bases to abelian schemes over a number field. His proof uses Hodge theory. We will give another approach that avoids Hodge theory but relies on higher dimensional Arakelov geometry. This will be the occasion to state some generalizations of diophantine statements in this theory. The contents of the talk will be based on joint work with Jean-Benoit Bost. 10.01.2012 Amador Martin-Pizarro (Lyon, z.Z. HU Berlin) On Schanuel's conjecture and CIT Abstract: Schanuel's conjecture states that given $n$ complex elements $(x_1,...,x_n)$ linearly independent over $\mathbb{Q}$, then $\mathrm{tr.deg} (x_1,...,x_n, \exp(x_1),...,\exp(x_n) )\geq n$. In recent research, Zilber has shown that there is a unique \emph{universal} field (up to isomorphism) of cardinality continuum equipped with a group homomorphism $\exp : \mathbb{G}_a \to \mathbb{G}_m$ where Schanuel's conjecture holds and such that the exponential-closure of a finite set is countable. The question remains whether $(\mathbb{C},\exp)$ is that field and in particular whether the key obstacle is Schanuel's conjecture itself. In order to prove some of the results, Zilber introduced a weak version of the \emph{Conjecture of Intersection with Tori}, which states that there is only finitely many cosets of tori (uniformely) for a given closed subvariety $V$ of $\mathbb{G}_m$ describing all possible atypical intersections of $V$ with any proper torus. In this talk, we will present some of the ideas in Zilber's work as well as an approach to weak CIT in positive characteristic by introducing Hasse-Schmidt iterative derivations in a separably closed field and relate the above to the existence of infinite Mersenne primes. 17.01.2012 Peter Toth (HU Berlin) A Geometric Proof of the Tamely Ramified Geometric Abelian Class Field Theory Abstract: Unramified geometric abelian class field theory establishes a connection between the Picard group and the abelianized etale fundamental group of a smooth, projective, geometrically irreducible curve over a finite field. We begin with a fairly detailed discussion of the unramified theory concentrating on Deligne's geometric proof. Then we turn to the tamely ramified theory, which transforms the classical situation to the open complement of a finite set of closed points of the curve, establishing a connection between a generalized Picard group and the tame fundamental group of the curve with respect to this finite set of closed points and present a geometric proof for the tamely ramified theory. 24.01.2012 Evelina Viada (Basel) Anomalous Varieties and the effective Mordell-Lang Conjecture Abstract: We consider an algebraic variety $V$ embedded in a product of elliptic curves $E^N$. It may happen that components of the intersection of $V$ with a proper algebraic subgroup of $E^N$ have dimension larger than expected. Such components are called $V$ anomalous varieties. The non density of all $V$ anomalous varieties in $V$, for all $V$ not contained in any algebraic subgroup, implies the Mordell-Lang Conjecture. Effective bounds for the height and degree of the maximal $V$ anomalous varieties gives the effective Mordell-Lang Conjecture. We will discuss these implications. We will give some new results for varieties of codimension 2 and some cases of the effective Mordell-Lang Conjecture for curves. 31.01.2012 Peter Scholze (Univ. Bonn) Die Hasse-Weil-Zeta-Funktion von Modulkurven Abstract: Ein alter Satz besagt, dass die Zeta-Funktion von Modulkurven geschrieben werden kann als Produkt von L-Funktionen von Modulformen und Hecke-Charakteren. Insbesondere folgt, dass sie eine meromorphe Fortsetzung hat und die erwartete Funktionalgleichung erfüllt. Der ursprüngliche Beweis beruht auf Arbeiten von Eichler, Shimura, Langlands, Deligne, und Carayol. Wir erklären die Methode von Langlands, und zeigen, wie diese erweitert werden kann, um an Primstellen schlechter Reduktion einen neuen, vereinfachten Beweis dieses Satzes zu liefern. 07.02.2012 Özlem Imamoglu (ETH Zürich) Periods of modular forms 14.02.2012 Alice Garbagnati (Mailand) Symplectic and non symplectic automorphisms of K3 surfaces Abstract: Fixed a particular family of K3 surfaces, the automorphisms group of a general member is very often unknown. In order to understand properties of the automorphisms groups of K3 surfaces it seems better to fix a particular group and to find the family of K3 surfaces admitting that group as subgroup of the full automorphisms group. The aim of this talk is to present some results on moduli spaces of K3 surfaces admitting a certain finite group $G$ as subgroup of the group of the automorphisms. We consider both groups acting symplectically on the surface (i.e. preserving the nowhere vanishing holomorphic two form) and group acting purely non symplectically (i.e. there is no elements in the group which preserve the nowhere vanishing holomorphic two form). One of the main results we present is that there exist some pairs of groups $(G,H)$ such that $G$ is a subgroup of $H$ and, under some hypothesis, a K3 surface $X$ admits $G$ as subgroup of the automorphisms group if and only if it admits $H$ as subgroup of the automorphisms group. This phenomenon happens in three distinct situations: both $G$ and $H$ act symplectically on the K3, both $G$ and $H$ act purely non symplectically on the K3, $G$ acts purely non symplectically and $H$ contains elements which are symplectic and elements which are non symplectic. Some of the results presented are obtained in collaboration with Alessandra Sarti.