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The Broadhurst-Kreimer conjecture and multiple zeta values
Broadhurst and Kreimer proposed a Poincare series for the algebra of multiple zeta values based on large scale numerical computations. In this talk, I will explain how the coefficients in their series are related to period polynomials and to structural properties of the formal multizeta value (double shuffle) Lie algebra.
Bounds for arithmetic intersection numbers
Arakelov geometry associates to an arithmetic surfaces an intrinsic invariant: the arithmetic self-intersection number of the dualizing sheaf. In this talk a result that implies upper bounds for this real number in particular for Fermat curves and modular curves will be presented.
Vector Bundles II
Some constructions of new groups and buildings
We give model theoretic constructions for a zoo of examples and counterexamples: we build groups and generalized n-gons satisfying strong transitivity properties. An ultraproduct of the latter yields a forest from which we obtain twin trees and homogeneous 1-round CAT(1)-spaces.
On Kac-Moody groups over rings
A split Kac-Moody group $G$ over a topological ring $R$ carries a natural topology defined by Kac and Peterson. In case the underlying topological ring $R$ is $k_\omega$, this turns the Kac-Moody group $G$ into a $k_\omega$ group, which allows for a certain amount of control over this topology. Each $\sigma$-compact locally compact ring is a $k_\omega$ ring, hence the above topology allows to study $S$-arithmetic subgroups of topological Kac-Moody groups over local fields. By using geometric and algebraic methods it is possible to observe Mostow-rigidity of $G(F_q[t^{-1}])$ in $G(F_q((t)))$ for sufficiently large $q$. In my talk I'd like to present the $k_\omega$ topology on $G$ and the Mostow-rigidity of $G(F_q[t^{-1}])$ in $G(F_q((t)))$. However, these methods fail to apply to the arithmetic subgroup $G(Z)$ of a real topological Kac-Moody group $G(\mathbb{R})$. I would be interested to know whether there exists a suitable concept of measure on Kac-Moody groups over local fields that would allow one to use geometric group theory in order to derive Mostow-rigidity uniformly as in the case of semisimple algebraic groups over local fields.
Fontaine-Lafaille theory II
Vector Bundles I
Moduli varieties of real and quaternionic bundles over a curve
We examine the moduli problem for real and quaternionic vector bundles over a curve, and we give a gauge-theoretic construction of would-be moduli varieties for such bundles. These moduli varieties are irreducible subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise, and we use this to study Gal(C/R)-actions on moduli varieties of semistable holomorphic bundles over a complex curve with a given real structure. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of those actions by $2^g +1$, where g is the genus of the curve. Moreover, we show that any two such connected components are homeomorphic.
Representations of braid groups and quantum groups
We construct some representations of braid groups, arising from the representations of quantum groups.
Fontaine-Lafaille theory
Self-corresponences of K3 surfaces via moduli of sheaves
In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See details in arXiv:0810.2945.
Galois representations
Galois representations and the Tame Inverse Galois problem
In this talk we address the following strengthening of the Inverse Galois problem over $\mathbb{Q}$, introduced by B. Birch around 1994: Let $G$ be a finite group. Is there a tamely ramified Galois extension of $\mathbb{Q}$ with Galois group $G$? When $G$ is a linear group, this problem can be approached through the study of Galois representations attached to arithmetic-geometric objects. Let $\ell$ be a prime number. We will consider the Galois representations attached to the $\ell$-torsion points of elliptic curves and abelian surfaces to give an explicit construction of tame Galois realizations of $GL(2, \ell)$ and $GSp(4,\ell)$.
Vector Bundles
Bending Fuchsian groups in complex hyperbolic space
In this talk, I will describe a family of embeddings of the Teichmüller space of a cusped Riemann surface Sigma in the $PU(2,1)$-representation variety of the fundamental group of Sigma. These embeddings are obtained by a bending process which allows an explicit description of an equivariant mapping from the universal cover of Sigma into the complex hyperbolic plane.
Isogenies of prime degree over number fields
Let K be a number field and E an elliptic curve defined over K. The so-called exceptional set for (E,K) consisting of prime numbers p such that E has a p-isogeny defined over K is finite iff E does not have CM over K. In this talk, I will state a criterion which allows, in various situations, to explicitly determine the exceptional set when it is finite.
Functional equations related to addition theorems
We discuss addition-theorems-type functional equations which naturally arise in various problems of mathematical physics and topology (Toda and Calogero-Moser dynamical systems, rational and pole solutions of the KdV equation; elliptic genera associated with the string-inspired Witten index).
Bloch's exact sequence for surfaces over local fields
Let $k$ be a local field such that $[k:Q_p] < \infty$ and $X$ be a proper smooth variety over $k$ with good reduction. Define $SK_1(X):=Coker(\partial: \bigoplus_{All C on X} K_2^M(k(C)) \to \bigoplus_{All points x on X} k(x)^*)$, where $\partial$ is the tame-symbol map. There is a reciprocity homomorphism $\rho_S:SK_1(S) \to \pi_1^{ab}(S)$ to the abelianized fundamental group of S. During my last stay in MPIM, I proved class field theory for S=elliptic fibration, by which I mean $\rho_S/m$ is bijective for any $m > 1$. Inspired by Faltings, I made much progress on my research thereafter and recently established that for any $m>1 \rho_S/m$ is bijective with $S$ an arbitrary surface, assuming good reduction. We mimic the original approach by S. Bloch who proved a certain Bloch's exact sequence to capture the size of $\pi_1^{ab}(S)$. The difference between Bloch's and ours is that we combine two semi-global theories successfully to deduce the global result. In the talk, I try to speak only high-lights and self-contained manner so that all audience enjoy what is the fascinating point in the result.
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English
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