Arithmetic Geometry week in Tokyo June 4 to 8, 2012 Department of Mathematical Sciences University of Tokyo, Japan Japanese version

Posters

Ramla Abdellatif (Paris-Sud) pdf
Title : About modulo p representations of p-adic reductive groups of rank 1
Abstract : Let G be a connected reductive group defined over a p-adic field F. The description of all the irreducible admissible smooth representations of G(F) having coefficients in an algebraically closed field of characteristic p is a hard problem, and most of the methods that have been developped in the complex setting completely fail in the mod p context. For now, such a description only exists for GL_2(Q_p) [Barthel-Livné, Breuil] and for SL_2(Q_p) [Abdellatif]. We present some results we proved when G is assumed to be quasi-split and of rank 1 over F. We give a special focus on the case G=SL_2, as we have more detailed statements in this setting, like a "mod p semi-simple Langlands correspondence".

Keisuke Arai (Tokyo Denki Univ.)
Title : Algebraic points on Shimura curves of Gamma_0(p)-type
Abstract : We classify the characters associated to algebraic points on Shimura curves of Gamma_0(p)-type, and over number fields (not only quadratic fields but also fields of higher degree) we show that there are few points on such a Shimura curve for every sufficiently large prime number p. This is an analogue of the study of rational points or points over quadratic fields on the modular curve X_0(p) by Mazur and Momose.

Patrick Forre (Tokyo),
Title : Higher Dimensional Class Field Theory of Varieties over Finite, Local and Higher Local Fields
Abstract : The goal of this work is to develop the higher dimensional class field theory of varieties over higher local fields. The main issue is the verification of a cohomological Hasse principle for regular models which are proper and flat over the corresponding complete discrete valuation ring. We are able to prove this in degree one, leading to an understanding of the reciprocity map of curves over higher local fields modulo n, where n is an integer prime to the final residue characteristic.

Hideaki Ikoma (Kyoto) pdf
Title : How to bound the successive minima on arithmetic varieties
Abstract : I would like to explain a new method to bound the last successive minima from above that are associated to high powers of a hermitian line bundle \overline{L} on an arithmetic variety X. As applications, we can prove the following results:
1) the last successive minima are generally bounded from above, provided that X is normal.
2) the sequence defining the sectional capacity of \overline{L} converge.
3) the sectional capacity of \overline{L} is Lipschitz continuous and birationally invariant.
4) necessary and sufficient conditions for H^0(X,mL) to have free basis consisting of strictly small sections for sufficiently large m.
5) a generalization of the theorem of successive minima of S. Zhang. In particular, we can reprove the general equidistribution theorem for rational points of small heights, which was first proved by Berman-Boucksom by using the Monge-Ampere operators.

Kensaku Kinjo (Tokyo, JSPS)
Title : Hypergeometric series and aritmetic-geometric mean over 2-adic fields
Abstract : Dwork proved that the Gaussian hypergeometric function on p-adic numbers can be extended to a function which takes values of the unit roots of ordinary elliptic curves over a finite field of characteristic p?3. We obtain an analogous theory in the case p=2. As an application, we give a relation between the canonical lift and the unit root of an elliptic curve over a finite field of characteristic 2 by the 2-adic arithmetic-geometric mean. This is a joint work with Y. Miyasaka.

Yuya Matsumoto (Tokyo) pdf
Title : On good reduction of some K3 surfaces
Abstract : Let X be a variety over a local field K. If X is an abelian variety, a theorem of Serre-Tate shows that X has good reduction if and only if its l-adic étale cohomology is unramified (a Galois representation of K is unramified if the action of the inertia group is trivial). I will prove that similar results hold if X belongs to certain classes of K3 surfaces.

Tomoki Mihara (Tokyo) pdf
Title : Analytic singular homology of non-Archimedean analytic spaces and Shnirel'man integral of differential forms along analytic cycles
Abstract : In non-Archimedean analysis, there is a classical integral theory called Shnirel'man integral, which is an analogue of the complex integral. The counterpart of the unit cycle S^1 is the group of roots of unit in the multiplicative group of an algebraic closure of the base field. Note that S^1 has two orientations, but there is no notion of orientation of the group of roots of unit. The lack of the orientation forces the integral to be normalised so that no counterpart of the period +-2\pi i appears. I generalised Shnirel'man integral as a integral of a differential form on an arbitrary analytic space over a local field with mixed characteristic, regarding a p-power root system of unit (= a group homomorphism from Z[p^-1] to the group of roots of unit) as an "oriented" 1-cycle. It canonically associates the period which is an element of Fontaine's p-adic period ring B_dR, and induces a B_dR-valued pairing between the space of holomorphic n-forms and the analytic singular homology obtained by the chain complex of paths defined in the similar way as the "oriented" 1-cycle. The pairing has the information of the periods of curves.

Kentaro Mitsui (Tokyo, JSPS) pdf
Title : Logarithmic transformations of rigid analytic elliptic surfaces
Abstract : We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic surfaces and non-algebraic rigid analytic surfaces.

Kentaro Nakamura (Sapporo) pdf
Title : Iwasawa theory of de Rham (phi,Gamma)-modules over the Robba ring
Abstract : By the results of Fontaine, Cherbonnier-Colmez and Kedlaya, the category of p-adic Galois representations of a p-adic field can be embedded in that of (phi,Gamma)-modules over the Robba ring. We develop the theory of Bloch-Kato's and Perrin-Riou's exponential maps in the framework of (phi,Gamma)-modules over the Robba ring. In particular, we generalize Perrin-Riou's map for all the de Rham (phi,Gamma)-modules.

Valentina Di Proietto (Tokyo, JSPS) pdf
Title : Kernel of the monodromy operator for semistable curves
Abstract : We reprove Chiarellotto's theorem that the invariant cycles for the monodromy acting on the de Rham cohomology of the generic fiber of a semistable curve are the rigid cohomology of the special fiber. This is done in an explicit way along the line of Coleman and Iovita's work. They are also able to define monodromy for coefficients: we extend such an invariant cycles sequence to the unipotent coefficients where we show that it is not always exact. We give an example of such a behavior together with a general condition for the non-exactness. This is a joint work with B. Chiarellotto, R. Coleman and A. Iovita.

Megumi Takata (Kyushu)
Title : Deligne's conjecture on the Lefschetz trace formula for p^n-torsion étale cohomology
Abstract : Deligne conjectured that, for any correspondence defined over a finite field with characteristic p, the Lefschetz trace formula holds with \ell-adic coefficients (\ell different from p) if we twist the correspondence by a sufficiently large power of the Frobenius endomorphism. This conjecture was proved by Kazuhiro Fujiwara under the most general situation. This formula is applied in the proofs of various Langlands conjectures. In this poster, I prove an analogous statement of Deligne's conjecture under one of the following three situations: (1) for automorphisms of finite order and p-torsion étale sheaves, (2) for smooth schemes and p-torsion étale sheaves, and (3) for proper smooth schemes and p^n-torsion étale sheaves.

Kazuki Tokimoto (Tokyo) pdf
Title : On the reduction modulo p of representations of a quaternion division algebra over a p-adic field
Abstract : The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases. We examine whether there exists a similar compatibility for the composition of the local Langlands correspondence and the local Jacquet- Langlands correspondence. The simplest case has already been considered by Vigneras. We deal with more cases.

Naoya Umezaki (Tokyo) pdf
Title : On uniform bound of the maximal subgroup of the inertia group acting unipotently on l-adic cohomology
Abstract : For a smooth projective variety over a local field, the action of the inertia group on the l-adic cohomology group is unipotent if it is restricted to some open subgroup. We give a uniform bound of the index of the maximal open subgroup satisfying this property. This bound depends only on the Betti numbers of X and certain Chern numbers of a projective embedding of X. This bound is independent of l.

The conference will take place in the conference hall of the Department of Mathematical Sciences, University of Tokyo, Komaba 153-8914 Tokyo, Japan.
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