Tag - Analytic number theory

Laurent Fargues: Locally symmetric spaces: p-adic aspects

30th November, 2017

p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. The proof consists in a thorough study of modifications of G-bundles on the curve. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction and other period spaces related to the basic locus in some Shimura varieties. This is joint work with Miaofen Chen and Xu Shen.

Hector Pasten: Shimura curves and new abc bounds

28th November, 2017

Existing unconditional progress on the abc conjecture and Szpiro's conjecture is rather limited and coming from essentially only two approaches: The theory of linear forms in p-adic logarithms, and bounds for the degree of modular parametrizations of elliptic curves by using congruences of modular forms. In this talk I will discuss a new approach as well as some unconditional results that it yields. For a fixed elliptic curve E over the rationals one has several modular parametrizations coming from various Shimura curves X, and our method amounts to using Arakelov theory to bound how these degrees vary as we change the source curve X, keeping E fixed. Unlike linear forms in p-adic logarithms, our method is global and deals with all local contributions at once. Concrete unconditional consequences will be discussed, such as bounding the number of divisors of abc triples polynomially on the radical, bounding the product of the ''fudge factors'' of elliptic curves polynomially on the conductor, and new lower bounds for truncated counting functions in the context of Vojta's arithmetic conjecture.

Ilya Khayutin: Joint equidistribution of CM points

21st November, 2017

A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity. The equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. I will explain the proof of a recent theorem making progress towards this conjecture.

Currently, this problem does not seem to be amenable to methods of automorphic forms even assuming GRH. Nevertheless, assuming a splitting condition at two primes the joining rigidity theorem of Einsiedler and Lindenstrauss applies. As a result the obstacle to proving equidistribution is the potential concentration of mass on graphs of Hecke correspondences and translates thereof. I will present a method to discard this possibility using a geometric expansion of a relative trace, description of the relative orbital integrals in terms of integral ideals and a sieve argument.

Francesc Castella: Elliptic curves of rank 2 and generalised Kato classes

24th October, 2017

The generalised Kato classes of Darmon-Rotger arise as p-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over ℚ of rank 2. In this talk, we will report on a joint work in progress with Ming-Lun Hsieh concerning a special case of the conjectures of Darmon-Rotger.

Alex Smith: 2∞-Selmer groups, 2∞-class groups, and Goldfeld’s conjecture

14th September, 2017

Take E/ℚ to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of E have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to 2k-Selmer groups for any k>1. In addition, using the close analogy between 2k-Selmer groups and 2k+1-class groups, we will prove that the 2k+1-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all k>1.

Olivier Fouquet: Congruences between motives and congruences between values of L-functions

13th April, 2017

If two motives are congruent, is it the case that the special values of their respective L-functions are congruent? More precisely, can the formula predicting special values of motivic L-functions be interpolated in p-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for p-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).