January 7 
NO SEMINAR 
January 14, 1011am 
Ana Caraiani (Princeton) I will explain a new construction and characterization of the padic local Langlands correspondence for GL_2(Q_p). This is joint work with Emerton, Gee, Geraghty, Paskunas and Shin and relies on the TaylorWiles patching method and on the notion of projective envelope. 
January 21 
Djordjo Milovic (ParisSud/Leiden) We will discuss some new density results about the 2primary part of class groups of quadratic number fields and how they fit into the framework on the CohenLenstra heuristics. Let Cl(D) denote the class group of the quadratic number field of discriminant D. The first result is that the density of the set of prime numbers p congruent to 1 mod 4 for which Cl(8p) has an element of order 16 is equal to 1/16. This is the first density result about the 16rank of class groups in a family of number fields. The second result is that in the set of fundamental discriminants of the form 4pq (resp. 8pq), where p == q == 1 mod 4 are prime numbers and for which Cl(4pq) (resp. Cl(8pq)) has 4rank equal to 2, the subset of those discriminants for which Cl(4pq) (resp. Cl(8pq)) has an element of order 8 has lower density at least 1/4 (resp. 1/8). We will briefly explain the ideas behind the proofs of these results and emphasize the role played by general bilinear sum estimates. 
January 28 
Ruochuan Liu (Beijing International Center for Mathematical Research) We construct a functor from the category of padic local systems on a smooth rigid analytic variety X over a padic field to the category of vector bundles with a connection on X, which can be regarded as a first step towards the soughtafter padic RiemannHilbert correspondence.As a consequence, we obtain the following rigidity theorem for padic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a padic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some results about the padic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. Joint work with Xinwen Zhu. 
February 4 
Annie Carter (UCSD)
JeanMarc Fontaine has shown that there exists an equivalence of categories between the category of continuous $\mathbb{Z}_p$representations of a given Galois group and the category of \'{e}tale $(\phi,\Gamma)$modules over a certain ring. We are interested in the question of whether there exists a theory of $(\phi,\Gamma)$modules for the LubinTate tower. We construct this tower via the rings $R_n$ which parametrize deformations of level $n$ of a given formal module. One can choose prime elements $\pi_n$ in each ring $R_n$ in a compatible way, and consider the tower of fields $(K'_n)_n$ obtained by localizing at $\pi_n$, completing, and passing to fraction fields. By taking the compositum $K_n = K_0 K'_n$ of each field with a certain unramified extension $K_0$ of the base field $K'_0$, one obtains a tower of fields $(K_n)_n$ which is strictly deeply ramified in the sense of Anthony Scholl. This is the first step towards showing that there exists a theory of $(\phi,\Gamma)$modules for this tower.

February 11 
NO MEETING 
February 18 
Maike Massierer (University of New South Wales)
Let C/Q be a curve of genus 3, given as a double cover of a conic with no Qrational points. Such a curve is hyperelliptic over the algebraic closure of Q but does not have a hyperelliptic model of the usual form over Q. We discuss an algorithm that computes the local zeta functions of C simultaneously at all primes of good reduction up to a given bound N in time (log N)^(4+o(1)) per prime on average. It works with the base change of C to a quadratic field K, which has a hyperelliptic model over K, and it uses a generalization of the "accumulating remainder tree" method to matrices over K. We briefly report on our implementation and its performance in comparison to previous implementations for the ordinary hyperelliptic case.

February 25 
Aly Deines (Center for Communications Research)

March 3 
NO MEETING 
March 10 
Francesc Fité (University of DuisburgEssen)
