L1

In this talk, I will describe a construction of a $p$-adic L-function attached to a triple of $p$-adic Coleman families of cusp forms.  This function interpolates algebraic parts of special values of Garrett triple product L-functions at balanced triples of weights.  Our construction is complementary to that of Harris and Tilouine which treats the case of unbalanced weights.

Given a $p$-adic weight and a finite slope we describe a Hecke and Galois equivariant geometric map relating elliptic overconvergent modular symbols and overconvergent modular forms of that slope, appropriate weights and $\mathbf{C}_p$-coefficients. We show that for a fixed slope, with the possible exception of a discrete family of weights, this map is an isomorphism.

Using Faltings' theory of the Hodge-Tate sequence of an abelian scheme we construct certain sheaves $\Omega^\kappa$, where $\kappa$ is a not-necessarily integral weight, over formal subschemes of modular varieties over which the canonical subgroup exists.   These sheaves generalize the integral powers, $\omega^k$, of the sheaf $\omega$ of relative differentials on a modular curve.   Global sections of $\Omega^\kappa$ provide geometric realizations of overconvergent automorphic forms of non-integral weight.  Applications of this approach to the theory of $p$-adic Hilbert modular forms will be given.   This is joint work with Fabrizio Andreotti and Adrian Iovita.

We apply the theory of the radius of convergence of a $p$-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact $p$-adic curves, smooth in the sense of rigid geometry. We show that a trivial lower bound for that radius implies a global form of Robert's $p$-adic Rolle theorem. The proof is based on a widely believed, although unpublished, result of simultaneous semistable reduction for finite morphisms of smooth $p$-adic curves. We also clarify the relation between the notion of radius of convergence used in our previous work and the more intrinsic one used by Kedlaya. (The paper is available athttp://arxiv.org/abs/1209.0081)

For a proper semistable curve over a DVR of mixed characteristics we re prove the ``invariant cycles theorem'' with trivial coefficients  by Chiarellotto i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber coincides with the first rigid cohomology group of the special fiber, without the hypothesis that the residue field is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber with coefficients convergent F-isocrystals given in a work of Coleman and Iovita. We apply these ideas to the case where the coefficients are unipotent convergent F-isocrystals defined on the special fiber: we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness. It is a joint work with B. Chiarellotto, R. Coleman and A. Iovita.

We extend the following theorem of H. Imai in several ways: If  $A$  is an abelian variety with potentially good reduction over a finite extension  $K$  of  $\mathbf{Q}_p$, then it has only finitely many rational torsion points over the maximal $p$-cyclotomic extension of  $K$. In particular, we prove the finiteness over $K(K^{1/p^\infty})$.

The overconvergence of the canonical subgroup of the universal abelian variety is one of the key ingredients of the theory of overconvergent modular forms. In this talk, I will show the overconvergence of the canonical subgroup with expected properties via the Breuil--Kisin classification, including the case of $p=2$.

In 1986 Mazur, Tate and Teitelbaum came up with a $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes $p$ of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.

I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti--Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure.
The key technique is the Taylor--–Wiles--–Kisin patching argument, which,when combined with a new, geometric perspective on the Breuil–--Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).

Wave-particle duality is a quantum behaviour usually assumed to have no possible counterpart in classical physics. We revisited this question when we found that a droplet bouncing on a vibrated bath could become self-propelled by its coupling to the surface waves it excites. A dynamical wave-particle association is thus formed.Through several experiments we addressed the same general question. How can a localized and discrete droplet have a common dynamics with a continuous and spatially extended wave? Surprisingly several quantum-like behaviors emerge; a form of uncertainty and a form of quantization are observed. I will show that both properties are related to the "path memory" contained in the wave field. The relation of this experiment with the pilot-wave models proposed by de Broglie and by Bohm for quantum mechanics will be discussed.