Journées de Géométrie Arithmétique de Rennes

6-10 Juillet 2009

Institut de Recherche Mathématique de Rennes

CNRS (UMR 6625), Université de Rennes 1, France

Organisateurs : Ahmed Abbes (CNRS, IRMAR, Rennes), Christophe Breuil (CNRS, IHES, Paris)
David Ellwood (Clay Mathematics Institute, Cambridge)
Mark Kisin (University of Chicago), Takeshi Saito (University of Tokyo)

### Présentations par affiches/Posters

Pour permettre une large contribution scientifique à la conférence, en particulier par les jeunes participants, il y aura une session de présentations par affiches. Les participants intéressés sont encouragés à soumettre des résumés à l'un des organisateurs.

To encourage more scientific contributions to the conference, especially from young participants, there will be a poster session. Interested participants are invited to submit abstracts for the poster session to one of the organizers.

K. Arai (Tokyo) Rational points on X_0^+(37M) (joint work with Fumiyuki Momose)

Résumé: One of the authors (Momose) studied rational points on the modular curve X_0^+(N) for a composite number N which has a prime divisor p different from 37 and the genus of X_0(p) is positive. The prime p=37 is peculiar because X_0(37) is a hyperelliptic curve and w_{37} is not the hyperelliptic involution. We show that the rational points on X_0^+(37M) consist of cusps and CM points. We also generalize the result for imaginary quadratic fields.

X. Caruso (Rennes) Dimensions de certaines variétés de Deligne-Lusztig affines généralisées

Résumé: Soit k un corps parfait de caractéristique p et sigma : k -> k une puissance (éventuellement négative) du Frobenius. On étend sigma en un endomorphisme d'anneaux de k((u)) en envoyant u sur u^b pour un certain entier positif b fixé. Étant donné un k((u))-espace vectoriel M muni d'une application sigma-semi-linéaire phi, on s'intéresse à l'espace de modules classifiant les k[[u]]-réseaux L de M pour lesquels la position relative de phi(L) par rapport à L est prescrite. Lorsque b = 1 et que sigma est non trivial, ces espaces de modules sont des variétés de Deligne-Lusztig affines pour le groupe GL_n, et sont maintenant assez bien compris. Ce n'est plus du tout le cas par contre si b > 1. Je donnerai dans ce poster plusieurs formules conjecturales pour estimer la dimension de ces espaces de module, ainsi que quelques résultats allant en faveur des conjectures. Un cas particulièrement intéressant est celui où b = p, car Kisin a montré récemment que les réseaux L que l'on classifie correspondent à des schémas en groupes sur l'anneau des entiers d'une extension totalement ramifiée de Frac W(k).

L. Fu (Nankai University) Global epsilon-factors for symmetric products of the Kloosterman sheaf

T. Hiranouchi (RIMS, Kyoto university) et Y. Taguchi (Kyushu University) Extensions of truncated discrete valuation rings

Résumé: A truncated discrete valuation ring is a commutative ring which is isomorphic to a quotient of finite length of a discrete valuation ring. We give an equivalence between the category of at most a-ramified finite separable extensions of a complete discrete valuation field K and the category of at most a-ramified finite extensions of the "length-a truncation" of the integer ring of K. This extends a theorem of Deligne, in which he proved this fact assuming the residue field is perfect. Our theory depends heavily on Abbes-Saito's ramification theory.

R. Liu (Jussieu) Locally analytic vectors of irreducible crystabelian representations of GL_2(Qp)

Résumé:Building on the recent works of Colmez and Berger-Breuil, we determine the locally analytic vectors of the unitary representations of GL_2(Qp) which correspond to irreducible crystabelian representations of G_{Qp} via the p-adic local Langlands correspondence of GL_2(Qp). This verifies a conjecture of Breuil.

K. Nakamura (Tokyo) Two dimensional trianguline representations of p-adic fields

Résumé: In Colmez's study of p-adic local Langlands correspondence for GL_2(Q_p), two dimensional trianguline representations play essential roles. In my poster, we give some results about two dimensional trianguline representations for general p-adic fields. For example, we give the complete classification of them and we prove Zariski density of them in deformation spaces of two dimensional representations.

A. Obus (University of Pennsylvania) Ramification of primes in fields of moduli of three-point covers

Résumé: A result of Beckmann states that for any three-point G-Galois cover of the Riemann sphere, if a prime p does not divide |G|, then p is unramified in the field of moduli of the cover. Wewers generalized this: if p exactly divides |G|, then p is tamely ramified in the field of moduli. We will discuss extensions of this result involving more general groups G with cyclic p-Sylow groups. For example, assume p is not 2. Then if G has a cyclic p-Sylow subgroup of order p^n and is p-solvable (a weaker condition than solvability), the nth higher ramification groups for the upper numbering above p vanish in the field of moduli. As in the case of Wewers' result, our proofs depend on a detailed analysis of the stable reduction of the G-Galois covers under consideration.

A. Pulita (Université de Montpellier 2) Differential equations over a 1-dimensional affinoid. p-adic deformation of differential equations: applications to the Morita p-adic Gamma function

Résumé: Let X be a 1-dimensional affinoid. Let $\sigma:X \to X$ be an automorphism of X. Let M be a differential module over X. We prove that if the automorphism $\sigma$ is sufficiently close to the identity, then there exists a canonical semilinear action of $\sigma$ on M. We construct then a functor, called deformation'', associating to a differential equation a module together with the semilinear action of $\sigma$ or a family $\Sigma$ of operators. We prove that this functor is fully faithful if the given family of operators respects a property of non degeneracy'' which is fulfilled in the most part of cases. The theorem can be generalized to open annuli, and more general rings of functions, in particular over the Robba ring. We apply then the above construction to tree cases: to the theory of (\phi,\Gamma)-modules, to the theory of q-difference equations, and finally we give an application to the Morita's p-adic Gamma function.

T. Tsushima (Tokyo) Jacobi sum and Hecke characters

Résumé: R. Coleman and W. MacCullum found a semi-stable model of the Fermat curve using rigid geometry and as a corollary they calculated the Jacobi sum and Hecke characters in 1988. In my poster, we give an elementary proof of the result of them on the calculation of the Jacobi sum and Hecke character without using rigid geometry.

G. Walker (Oxford) Minimalism on Elliptic Curves and Computing Ranks using p-adic Cohomology

Résumé: One application of recently developed algorithms in rigid and crystalline cohomology is computing ranks of the Mordell-Weil group of Jacobians of low genus curves defined over function fields of positive characteristic. We show how one can remove the necessity of a 'fuchsian basis' for the Picard-Fuchs differential system, and hence compute L-functions of large numbers of 'general' curves. Data gathered using this algorithm shall be presented, supporting for the first time the 'minimalism conjecture' on elliptic curves, that is, one half of all elliptic curves have infinitely many points.

L. Xiao (MIT) Non-archimedean differential modules and ramification theory

Résumé: The theory of non-archimedean differential modules has many applications to the ramification theory. In particular, one can deduce the fundamental Hasse-Arf theorem for the arithmetic ramification filtrations using this theory as a main tool. We also obtain certain variational property of the Swan conductors. This leads to understanding global ramification for varieties.

M. Yoshida (Kyushu) Ramification of local fields and Fontaine's property (Pm)

Résumé: Let K be a complete discrete valuation field with perfect residue field. Consider the ramification filtration (G^j) in the upper numbering of a finite Galois extension L of K. Then Fontaine characterized the greatest break of the ramification filtration by a certain property (Pm) of the extension L/K for real numbers m. By refining Fontaine's result, we obtain a new interpretation of the ramification filtration in terms of the property (Pm).