All Souls College
Oxford OX1 4AL
United Kingdom
Talk: “Algebraic classes in mixed characteristic and Andre's p-adic periods”
(Joint work with D. Fratila) Motivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of ℚ_{p} which we call the algebra of “Andre's p-adic periods”. We will explain the analogy and the difference between these p-adic periods and the classical complex periods. For instance, they both contain several examples of special values of classical functions (logarithm, gamma function, …) and they share transcendence properties. On the other hand, the classical Tannakian formalism which is used to bound the transcendence degree of complex periods has to be modified in order to be used in the p-adic setting. We will discuss concrete examples of all these instances though elliptic curves and Kummer extensions.
Talk: “Formal-analytic geometry, degree bounds and Lefschetz-Nori theorems on fundamental groups”
We will introduce the notion of a formal-analytic arithmetic surface, which encapsulates the setting of classical algebraization theorems on power series and explain how their study is amenable to geometric techniques. We will provide applications to finiteness results on fundamental groups that provide arithmetic analogues of classical results of Nori in complex-analytic geometry.
Talk: “Canonical lifts of logarithmic differential forms on the universal vector extension”
(Joint work in progress with Nils Matthes). The universal vector extension of an elliptic curve is a 2-dimensional commutative algebraic group satisfying remarkable geometric properties. In particular, in characteristic zero, it is an affine bundle over the elliptic curve over which every unipotent vector bundle becomes canonically trivial. This property allows us to describe the unipotent de Rham fundamental group of punctured elliptic curves in terms of globally defined logarithmic differential forms, much like the genus 0 case. This talk concerns families of elliptic curves. I will explain how the so-called "crystalline nature" of the universal vector extension gives canonical lifts of logarithmic differential forms relative to some base scheme to absolute logarithmic forms. As an application, this leads to a purely algebraic construction of the universal elliptic KZB equations in arbitrary levels.
Talk: “The ubiquity of Sidon sets in tannakian computations”
A Sidon set is a subset S of an abelian group G with the property that an element of G cannot be the sum of two elements of S in more than one way. I will give examples of Sidon sets coming from algebraic geometry and explain the rôle they play in computations of tannakian fundamental groups in various settings, among which exponential motives and Fourier transforms of trace functions on algebraic groups over finite fields. The talk is based on joint work with A. Forey and E. Kowalski on one hand, and P. Jossen on the other hand.
Talk: “Central L-values of elliptic curves and their p-adic valuation”
In this talk, I will study elliptic curves E of the form x^{3}+y^{3}=N for positive integers N. They admit complex multiplication, which allows us to tackle the conjecture of Birch and Swinnerton-Dyer for E effectively. Indeed, using Iwasawa theory, Rubin was able to show the p-part of the conjecture for E for all primes p, except for the primes 2 and 3. The theory becomes much more complex at these small primes, but at the same time we can observe some interesting phenomenons. I will explain a method to study the p-adic valuation of the algebraic part of the central L-value of E, and I will establish the 3-part of the conjecture for E in special cases. I will then explain a relation between the 2-part of a certain ideal class group and the Tate-Shafarevich group of E. Part of this talk is based on joint work with Yongxiong Li.
Talk: “Limits of Mahler measures and exactness of polynomials” (slides)
Algebraic relations between periods are predicted by Grothendieck's period conjecture. But how should one study their topology? In the first part of my talk, based on joint work with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei, I will show how one can produce some very interesting Cauchy sequences of periods. This is done by looking at the Mahler measure of a polynomial, an invariant introduced in 1962 in transcendental number theory. We are able moreover to give an explicit bound for the error term in the convergence of these sequences, and a full asymptotic exapansion for one explicit family of polynomials. The Mahler measure of these polynomials can be expressed in terms of the Bloch-Wigner dilogarithm evaluated at certain roots of unity. This is due to the fact that these polynomials share the property of being exact, which was introduced in the work of Maillot and Lalin. In the second part of my talk, based on work in progress with François Brunault, I will introduce this notion briefly, and a generalization of it, and their utilities in predicting links between Mahler measures and special values of L-functions.
Talk: “Algebraicity of critical Hecke L-values” (slides)
Euler's beautiful formula on the values of the Riemann zeta function at the positive even integers can be seen as the starting point of the investigation of special values of L-functions. In particular, Euler's result shows that all critical zeta values are rational up to multiplication with a particular period, here the period is a power of 2πi. Conjecturally this is expected to hold for all critical L-values of motives. In this talk, I will explain a joint result with Guido Kings on the algebraicity of critical Hecke L-values up to explicit periods for totally imaginary fields.
Talk: “Frobenius structure and p-adic zeta values”
For an ordinary linear differential equation a p-adic Frobenius structure is an equivalence between the local system of its solutions and its pullback under the map t→t^{p} which is realized over the field of p-adic analytic elements. Its existence is a strong property, we only expect it for differential equations arising from the Gauss-Manin connection in algebraic geometry. In a vicinity of a singular point such a structure can be described by a bunch of p-adic constants. We will show examples of families of hypersurfaces for which these constants turn out to be p-adic zeta values. This is joint work with Frits Beukers.
Time | Monday 27th | Tuesday 28th |
---|---|---|
9.30–10.30 | Talk Pengo | Talk Fonseca |
coffee break | coffee break | |
11.30–12.30 | Talk Sprang | Talk Charles |
lunch in the Wharton room | ||
14.30–15.30 | Talk Fresan | Talk Ancona |
coffee break | coffee break | |
16.00–17.00 | Talk Kezuka |