Speaker: Bryden Cais (McGill)

Title: de Rham cohomology and p-adic analytic families of ordinary Galois representations.

Abstract: In the 1980's, Hida constructed certain p-adic analytic families of ordinary Galois representations via a detailed study of Hecke algebras and group cohomology. Shortly after this, Mazur and Wiles gave a geometric interpretation of the associated families of Galois representations by realizing them in the etale cohomology groups of towers of modular curves. In accordance with the philosophy of p-adic Hodge theory, one expects that there should be a corresponding geometric construction of p-adic families of ordinary modular forms via de Rham cohomology. In this talk, we will explain such a construction; as a consequence, we obtain a new and purely geometric approach to Hida theory. Using work of Ohta, we will elucidate how our construction relates to that of Mazur-Wiles via standard comparison isomorphisms in p-adic Hodge theory.

Speaker: Olivier Fouquet (Osaka)

Title: Control theorems for Selmer groups of nearly ordinary deformations.

Abstract: Starting with the influential work of B.Mazur, it has been noticed that the Selmer group of a Galois deformation specialized at an arithmetic point is almost isomorphic to the Selmer group of the specialization. This control theorem reflects the interpolation property of the p-adic L-function. In a joint work with T.Ochiai, we prove a control theorem for the Selmer group of nearly ordinary deformations of representations of the Galois group of a totally real field.

Speaker: Chung Pang Mok (Berkeley)

Title: Heegner points and p-adic L-functions for elliptic curves over certain totally real fields.

Abstract: For an elliptic curve E over Q satisfying suitable hypothesis, Bertolini and Darmon have derived a formula for the Heegner point on E, in terms of a certain central derivative of the two variable p-adic L-function associated to E. We give a partial generalization of their work to the setting of totally real fields. This has application to special values of L function of elliptic curves.

Speaker: Jeehoon Park (McGill)

Title: Plus/minus p-adic L-functions for Hilbert Modular forms.

Abstract: When p is an ordinary prime, then the p-adic L-function of a modular form is an Iwasawa function (coming from p-adic measure). But if p is supersingular, p-adic L-functions are no longer Iwasawa functions. R. Pollack realized when the Hecke eigen value a_p of an elliptic modular form is zero (a particular supersingular case) , there exists a plus/minus p-adic L-function which comes from a p-adic measure. The goal of this talk is generalizing his result to Hilbert modular form case. This is joint work with Shahab Shahabi.

Speaker: Jonathan Pottharst (Boston College)

Title: Generalizations of Greenberg's "ordinary" condition.

Abstract: We present various weakenings of Greenberg's "ordinary" hypothesis for a p-adic Galois representation, and extend to them important properties, such as (potential) semistability and computation of the Bloch--Kato local condition. Time permitting, we explain how our notions fit into a conjectural framework in which to construct rather general Galois-theoretic eigenvarieties.

Speaker: Tomoyuki Abe (Tokyo)

Title: Sheaf of microdifferential operators and its application to the characteristic varieties on curves.

Abstract: A. Marmora defined the sheaves of microdifferential operators of level 0 on curves. In this talk, first, we define sheaves of microdifferential operators of arbitrary levels. A difficulty of dealing with these sheaves is that there are no homomorphisms between sheaves of microdifferential operators of different levels. To remedy this situation, we define the intermediate differential operators, and using this, we define the sheaf microdifferential operators for $\mathscr{D}^\dag$. Finally, we prove that the characteristic variety and the support of microlocalization of a $\mathscr{D}^\dag$-module are the same in the curve case.

Speaker: Kentaro Nakamura (Tokyo/Keio)

Title: Zariski density of two dimensional trianguline representations of p-adic fields.

Abstract: TBA

Speaker: Kensaku Kinjyo (Tohoku)

Title: 2-adic arithmetic-geometric mean and elliptic curves.

Abstract: Henniart and Mestre defined the arithmetic-geometric mean over a p-adic field for all primes p, and related it with an elliptic curve having multiplicative reduction over the p-adic field. Unfortunately, unless p = 2, their method cannot be generalized to elliptic curves with good reduction, essentially because the isogeny which connects the arithmetic-geometric mean with elliptic curves is of degree 2. In this talk, we report that the sequence of the ratio of 2-adic arithmetic-geometric mean sequences converges periodically. Moreover, using these convergence, we obtain the canonical lift of the reduction of an elliptic curve having good ordinary reduction over the 2-adic field.

Speaker: Yuken Miyasaka (Tohoku)

Title: p-adic arithmetic-geometric mean and periods of a Tate curve.

Abstract: Gauss studied the arithmtic-geometric mean and discovered a connection between it and the periods of an elliptic curve de?ned over the real number ?eld. In 1989 Henniart and Mestre de?ned the p-adic arithmetic- geometric mean and related it with the period of a Tate curve. In this talk, we will give a relation between the p-adic arithmetic-geometric mean and Hodge- Tate periods of the Tate curve.

Speaker: Kenichi Namikawa (Osaka)

Title: On mod p non-vanishing of special values of L-functions associated to modular forms over imaginary quadratic fields.

Abstract: Let $f$ be a cusp form of $GL(2)$ over rational number field and we take an arbitrary prime $p$ independently of $f$. There exists a complex number called complex period of $f$ which is determined up to $p$-adic unit. Then it's known by Shimura that a ratio of the period and the special value of twist of $L$ function of $f$ by an arbitrary Dirichlet character is an algebraic number. Furthermore, Ash, Stevens and Prasanna proved that there exists a Dirichlet character such that the ratio is actually a $p$-adic unit. In this talk, we show an analogous result of the latter for cusp forms of $GL(2)$ over imaginary quadratic fields.

Speaker: Masataka Chida (Kyoto)

Title: Heegner cycles and the central value of L-functions for modular forms

Abstract: Following Bloch-Kato's Tamagawa number conjecture, it is expected that the critical central value of L-functions for modular forms relates the size of Selmer groups for the p-adic Galois representations associated to modular forms. In this talk, I will explain an approach to this conjecture using Heegner cycles on Kuga-Sato varieties over a Shimura curve.

Speaker: Tetsushi Ito (Kyoto)

Title: On the weak Lefschetz theorem for singular varieties and the supersingular strata of Siegel threefolds

Abstract: It is well-known that the weak Lefschetz theorem for etale cohomology claims the k-th degree etale cohomology of a projective smooth variety X of dimension n coincides with the cohomology of a smooth projective hypersurface H if the inequality "k < n" holds. This theorem does not hold in general if X has singularities. We give a version of the weak Lefschetz theorem for singular varieties for H^1 under some conditions, and its application to the vanishing of the cohomology of Siegel threefolds. Geometry of minimal/toroidal compactification and the supersingular strata are used in the proof.