Résumés

Boris Adamczewski (CNRS, Lyon), E- et M-fonctions : concordance et divergence

La transcendance et l'indépendance algébrique des valeurs des E-fonctions de Siegel et des M-fonctions de Mahler en des points algébriques est un problème étudié depuis la fin des années 1920. De façon assez mystérieuse, ces deux classes de fonctions analytiques de nature très différente donnent aujourd'hui lieu à deux théories transcendantes jumelles. Dans cet exposé, je passerai en revue les principaux résultats qui fondent ces théories en insistant sur leurs conséquences et aussi leurs limites. J'évoquerai également un point important de divergence révélé par un résultat plus récent obtenu en collaboration avec Colin Faverjon.

Yves André (CNRS, Paris), « Ein Bourgeois, wer noch Algebra treibt! Es lebe die unbeschränkte Individualität der transzendenten Zahlen! » (C.-L. Siegel)

Je présenterai un panorama des développements de la première partie de l’article fondateur de Siegel, mettant l‘accent sur les G-fonctions/opérateurs plutôt que sur les E-fonctions/opérateurs (leurs transformées de Fourier, qui seront l’objet d’autres exposés); sur leur rapport aux motifs et à la géométrie diophantienne des intersections improbables etc.

Frits Beukers (Utrecht), Introduction to E-functions and their values

In this lecture we give an introduction to E-functions and arithmetic properties of their values at algebraic arguments. Although the material is known to the experts, I hope to include some modest new result.

Mireille Bousquet-Mélou (CNRS, Bordeaux), Diagonals of rational functions in enumerative combinatorics

Given a class A of objects equipped with a size, the (ordinary) generating function of A is the formal power series ∑_n a(n)tⁿ, where a(n) is the number of objects of A of size n. We will survey classical families of D-finite series (rational, algebraic, diagonal of rational) and discuss what it means for a class A to have its generating function in this family; and, most importantly, how this can be proved or disproved.

Xavier Caruso (CNRS, Bordeaux), Viewpoints on the p-curvature

This talk will serve as a gentle introduction to the p-curvature, a notion of first importance in the fascinating world of linear differential equations in positive characteristic. We shall give different definitions of the p-curvature, each of them illuminating different facets of this intricate object. In particular, we will explain how it controls the existence of solutions both in characteristic p and in characteristic zero. A special attention will be given to examples and effective computations of the p-curvature.

Eric Delaygue (Lyon), A Lindemann-Weierstrass theorem for E-functions

After discussing the recent effective results on the exceptional values of E-functions, I will show that Beukers’ lifting result and André’s theory of E-operators lead to a Lindemann-Weierstrass theorem for E-functions. Then I will present few applications, in particular I will show that the transcendental values at algebraic arguments of a hypergeometric E-function are linearly independent over the field of algebraic numbers.

Clément Dupont (Montpellier), Motivic Galois theory for algebraic Mellin transforms 

This talk will discuss series expansions of algebraic Mellin transforms, and the periods that appear as their coefficients. The basic example is Euler's beta function, whose series expansion features values of the Riemann zeta function at integers. I will explain how the motivic Galois group acts on series expansions of algebraic Mellin transforms, and give examples. As an application, we obtain a ''cosmic Galois theory'' (prophesized by Cartier) for Feynman integrals in dimensional regularization. This is joint work with Francis Brown, Javier Fresán, and Matija Tapušković. 

Stéphane Fischler (Orsay), Rings of values at algebraic points of G-functions and E-functions

In this lecture (devoted to joint works with Tanguy Rivoal) we introduce G- and E-functions, defined by Siegel in 1929. These are special functions of which the Taylor coefficients at 0 are algebraic, and have good arithmetic properties. Results of André, Chudnovsky and Katz prove that they are annihilated by specific differential operators. Of great interest to us are the values taken by these functions at algebraic points: they are closely related to periods and exponential periods. To conclude, asymptotic expansions of E-functions will be mentioned. They make it possible to prove, assuming standard conjectures on exponential periods and motivic Galois groups, that there exists an E-function which is not a polynomial in hypergeometric E-functions. This is a negative answer to a question asked by Siegel; the same result has been proved recently, unconditionaly, by Fresán and Jossen. 

Javier Fresán (École polytechnique), Periods and special values of G-functions

I will explain the equivalence between three different presentations of the ring of periods and how one of them, due to Ayoub, allows one to prove that every period is a special value of a G-function. The converse statement is an avatar of the Bombieri-Dwork conjecture. If time permits, I will then discuss what can be expected for exponential periods and special values of E-function, as a warm-up for Peter Jossen's talk.

Peter Jossen (Londres), Siegel's question on hypergeometric E-functions

Right after introducing the notion of E-function, Siegel proved as an example that certain hypergeometric series are E-functions. Then he asked the question whether every E-function can be written as a polynomial expression in such hypergeometric series. I will explain how to build E-functions out of geometric data and how to produce an invariant that distinguishes those of hypergeometric origin. This will lead us to the conclusion that, among all E-functions, those of hypergeometric origin form a very very very very very very small subset. Joint work with Javier Fresán.    

Frédéric Jouhet (Lyon), Cyclotomic valuations of q-Pochhammer symbols and q-integrality of basic hypergeometric series

Non trivial arithmetic properties for factorial ratios and their generating series can be derived from an elementary step function, the Landau delta function. Such results can non only be extended to classical q-analogs of these ratios, but also to ratios of Pochhammer symbols appearing in the generalized hypergeometric series. However for the latter, one needs more complicated arithmetical functions, due to Dwork and Christol. We will explain how to generalize this to ratios of q-analogs of Pochhammer symbols appearing in the basic hypergeomtric series. It will enable us to give on the one hand the cyclotomic valuation of q-Pochhammer symbols, and on the other hand a criterion for q-integrality of their ratios. These provide suitable q-analogs of two results due to Christol: a formula for the p-adic valuation of Pochhammer symbols and a criterion for the N-integrality of hypergeometric series. This is joint work with Boris Adamczewski, Jason Bell, and Eric Delaygue.

Pierre Lairez (INRIA, Saclay), The 22 periods of a quartic surface and their integer relations

A smooth complex quartic surface in the projective 3-space defines 22 complex numbers called periods. The integer relations between them describe the algebraic curves lying on this surface. I will explain how to compute numerically the periods with high precision and recover heuristically the lattice of integer relations. In a second part, I will show some options to certify the computation, especially separation bounds. This is joint work with Emre Sertöz (Leibniz University Hannover).

Julien Roques (Lyon), E-functions of order 2 and rigidity

We will explain how, using the structure of E-operators and rigidity, we can obtain an alternative proof of a result of Gorelov stating that the E-functions of order at most 2 are essentially hypergeometric. This is a joint work with T. Rivoal.

Bruno Salvy (INRIA, Lyon), Minimization of differential equations and algebraic values of E-functions

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel’s E-functions take algebraic values. We present algorithms for these questions and discuss implementation and experiments. Joint work with Alin Bostan and Tanguy Rivoal.

Masha Vlasenko (IMPAN, Varsovie), Integrality of instanton numbers

This is joint work with Frits Beukers. In early 1990s physicists predicted that numbers of rational curves of fixed degree on the generic quintic threefold are equal to the so called instanton numbers, which are calculated in terms of solutions of a differential equation on its mirror manifold. By construction these numbers are rational. In turns out that p-integrality of instanton numbers follows from vanishing of a certain p-adic constant defined in terms of the p-adic Frobenius structure on the differential equation. I will introduce the necessary ingredients and show that the p-adic constants of our interest can be computed using congruences for expansion coefficients of differential forms. This approach yields a proof of p-integrality for almost all p in several key examples of mirror symmetry.