I will show that Fontaine's ring of p-adic periods can be realized as the ring of universal p-adic constants in the sense of derived algebraic geometry, and discuss a possible new construction of the p-adic periods map.

Let $F$ be a perfect field of characteristic $p>0$ equipped with a non trivial absolute value, $E$ a non archimedean locally compact field whose residue field is contained in $F$ and $pi$ a uniformizing parameter of $E$. We associate functorially to these datas a separated integral noetherian regular scheme $X=X_{F,E,pi}$ of dimension $1$ defined over $E$. There is an equivalence of categories between semi-stable vector bundles of slope $0$ over $X$ and continuous $E$-linear representations of the absolute Galois group $H_F$ of $F$.

When $F$ is algebraically closed, the closed points of $F$ can be described in terms of the Lubin-Tate formal group of $E$ corresponding to $pi$.

If $C$ is the $p$-adic completion of $overline Q_p$, one can associate to $C$ an algebraically closed field $F=F(C)$ as above and ${rm Gal)(overlineQ_p/Q_p)$ acts on the curve $X=X_{F(C),Q_p,p}$. The two main results of $p$-adic Hodge theory can be recovered from the classification of vector bundles over $X$.

(joint work with Laurent Fargues)

Read more at http://www.math.u-psud.fr/~fargues/Prepublications.html.

I will explain some results towards the Langlands-Rapoport conjecture which predicts the structure of the mod p points of a Shimura variety. A consequence of the conjecture is that the isogeny class of every mod p point contains a point which admits a lifting to a special (ie CM) point of the Shimura variety. One of the roots of the subject is the work of John Tate on CM liftings and endomorphisms of abelian varieties mod p.

In the past three decades there have been some exciting applications of elliptic curves over finite fields to integer factoring, primality testing, and cryptography. These applications in turn have raised some interesting problems often of an unconventional flavor. For example, how often is the order of an elliptic curve group prime, or how often does it have all small prime factors? In this talk we will visit problems such as these, as well as other analytic-type problems relating to ranks of elliptic curves over function fields and to elliptic divisibility sequences.

We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p.  By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar.  The only distribution on Mordell-Weil ranks compatible with both our random model and Delaunay's heuristics for Sha[p] is the distribution in which 50% of elliptic curves have rank 0, and 50% have rank 1.  We generalize many of our results to abelian varieties over global fields.  This is joint work with Eric Rains.

I will discuss the statement and proof of an Equivariant Main Conjecture (EMC) in the Iwasawa theory of arbitrary global fields. This will be followed by applications of the EMC (via Iwasawa co-descent) towards proving various well known conjectures on special values of global L-functions. In the process, an important role will be played by an explicit construction of ell-adic Tate sequences. This is based on joint work with Cornelius Greither (Munich).

If we specialize algebraic equations having good properties, we usually face degeneracies. Starting with a bad specialization, we can try to improve it , performing modifications under control. If we succeed to get a new specialization with the initial good properties preserved,we get a permanence statement. We shall present examples of permanence with particular interest concerning semi-stable models.

We know, thanks to the Weil conjectures, that counting points of varieties over finite fields yields purely topological information about them. In this talk I will first describe how we may count the number of points over finite fields on the character varieties parameterizing certain representations of the fundamental group of a Riemann surface into GL_n. The calculation involves an array of techniques from combinatorics to the representation theory of finite groups of Lie type. I will then discuss the geometric implications of this computation and the conjectures it has led to.

This is joint work with T. Hausel and E. Letellier

 In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields.  We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) with a given 2-Selmer rank.  As a consequence, under appropriate hypotheses there are many twists with Mordell-Weil rank zero, and (assuming the Shafarevich-Tate conjecture) many others with Mordell-Weil rank one.  Another application of our methods, using ideas of Poonen and Shlapentokh, is that if the Shafarevich-Tate conjecture holds then Hilbert's 10th problem has a negative answer over the ring of integers of any number field.