# 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

 January 7 NO SEMINAR 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 NO MEETING 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) Elliptic Curve Parameterizations by Modular curves and Shimura curves A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, there is a non-constant map from the modular curve $X_0(N)$ to $E$. For some curve isogenous to $E$, the degree of this map will be minimal; this is the modular degree. The Jacquet-Langlands correspondence allows us to similarly parameterize elliptic curves by Shimura curves. In this case we have several different Shimura curve parameterizations for a given isogeny class. Further, this generalizes to elliptic curves over totally real number fields. In this talk I will discuss these degrees and I compare them with $D$-new modular degrees and $D$-new congruence primes. This data indicates that there is a strong relationship between Shimura degrees and new modular degrees and congruence primes. March 3 NO MEETING March 10 Francesc Fité (University of Duisburg-Essen) Fields of definition of CM elliptic k-curves and Sato-Tate groups of abelian surfaces Let A be an abelian variety defined over a number field k that is isogenous over an algebraic closure to the power of an elliptic curve E. If E does not have CM, by results of Ribet and Elkies concerning fields of definition of k-curves, E is isogenous to an elliptic curve defined over a polyquadratic extension of k. We show that one can adapt Ribet's methods to study the field of definition of E up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato-Tate groups of abelian surfaces: First, we show that 18 of the 34 possible Sato-Tate groups of abelian surfaces over Q, only occur among at most 51 Qbar-isogeny classes of abelian surfaces over Q; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato-Tate groups of abelian surfaces. This is a joint work with Xevi Guitart. Preparatory talk: In the preparatory talk I plan to review very briefly basic definitions concerning abelian varieties necessary to introduce (in the main talk) the notion of abelian k-variety. I will also present the (general) Sato-Tate conjecture and show how it motivates the problem considered in the main talk.