UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421

This quarter, most talks will be preceded by a 30-minute "prep talk" for graduate students and postdocs. These will be in AP&M 7421 starting at 1:15pm.

Winter Quarter 2016

For previous quarters' schedule, click here.

January 7


January 14, 10-11am
(note unusual time)

Ana Caraiani (Princeton)
Patching and the p-adic local Langlands program for GL_2(Q_p)

I will explain a new construction and characterization of the p-adic local Langlands correspondence for GL_2(Q_p). This is joint work with Emerton, Gee, Geraghty, Paskunas and Shin and relies on the Taylor-Wiles patching method and on the notion of projective envelope.

January 21
(plus prep talk)

Djordjo Milovic (Paris-Sud/Leiden)
Density results on the 2-part of class groups

We will discuss some new density results about the 2-primary part of class groups of quadratic number fields and how they fit into the framework on the Cohen-Lenstra 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 16-rank 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 4-rank 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
(plus prep talk)

Ruochuan Liu (Beijing International Center for Mathematical Research)
Rigidity and Riemann-Hilbert correspondence for de Rham local systems

We construct a functor from the category of p-adic local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with a connection on X, which can be regarded as a first step towards the sought-after p-adic Riemann-Hilbert correspondence.As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic 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 p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. Joint work with Xinwen Zhu.

February 4
(plus prep talk)

Annie Carter (UCSD)
Lubin-Tate Deformation Spaces and $(\phi,\Gamma)$-Modules

Jean-Marc 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 Lubin-Tate 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.
In this talk we will introduce the notions of formal modules and their deformations, strictly deeply ramified towers of fields, and $(\phi,\Gamma)$-modules, and sketch the proof that the Lubin-Tate tower is strictly deeply ramified.

February 11


February 18
(plus prep talk)

Maike Massierer (University of New South Wales)
Counting points on some geometrically hyperelliptic curves of genus 3 in average polynomial time

Let C/Q be a curve of genus 3, given as a double cover of a conic with no Q-rational 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.

Joint work with David Harvey and Andrew V. Sutherland.

In the pre-talk, we will introduce some of the objects that the talk is about, such as curves and their models, the zeta function and how it relates to point counting, and the particular type of genus 3 curves that we are interested in. Counting points on some geometrically hyperelliptic curves of genus 3 in average polynomial time.

February 25
(plus prep talk)

Aly Deines (Center for Communications Research)

March 3


March 10

Francesc Fité (University of Duisburg-Essen)