January 7 |
NO SEMINAR |
January 14, 10-11am |
Ana Caraiani (Princeton) 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 |
Djordjo Milovic (Paris-Sud/Leiden) 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 |
Ruochuan Liu (Beijing International Center for Mathematical Research) 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 |
Annie Carter (UCSD)
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.
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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 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.
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February 25 |
Aly Deines (Center for Communications Research) 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)
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.
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