E-functions and geometry

28 September - 21 December 2023

Thursdays, 10:15 - 12:00

Location: HG G 43

First lecture: 28 September

Abstract

E-functions were introduced by Siegel in a landmark 1929 paper [4] with the goal of generalizing to other special functions the transcendence results for the values of the exponential function at algebraic numbers by Hermite, Lindemann, and Weierstrass. E-functions are power series with algebraic coefficients that are solutions of a linear ordinary differential equation with polynomial coefficients, and whose Taylor coefficients satisfy a growth condition of arithmetic nature. Besides the exponential, examples include Bessel functions and
a rich family of hypergeometric series. Among their remarkable properties is the fact that, according to the Siegel–Shidlovsky theorem, all algebraic relations between special values of E-functions arise by specialization from functional relations.

The study of E-functions has expanded considerably over the last twenty years, starting from a seminal work of Y. André [1] which determines the structure of the differential equations they satisfy. More recently, the links with arithmetic geometry and especially the theory of exponential periods [3] have shed new light on the geometric origin of E-functions, resulting for instance in the solution of a long-standing problem by Siegel on the existence of non-hypergeometric E-functions [2]. Many mysteries remain, however.

The goal of this Nachdiplomvorlesung will be to present in a systematic and accessible manner the modern theory of E-functions, its applications, and open problems.

Tentative outline
(1) E-functions and G-functions: definitions and examples.
(2) The Siegel–Shidlovsky Theorem. The proof by André–Beukers.
(3) Structure of the differential equations satisfied by E-functions.
(4) An introduction to exponential motives.
(5) Exponential period functions are E-functions.
(6) Speculations about the arithmetic of series like\(\sum_{n\geq 0} n!z^n\).

References
[1] Y. André,Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, II. Transcendance sans transcendance, Ann. of Math. (2)151no. 2 (2000), 705–756.
[2] J. Fresán and P. Jossen,non-hypergeometric E-function, Annals of Math.194(2021), 903–942.
[3] J. Fresán and P. Jossen,Exponential motives, preprint available atexternal pagehttp://javier.fresan.perso.math.cnrs.fr/expmot.pdfcall_made.
[4] C. L. Siegel,Über einige Anwendungen diophantischer Approximationen, Abhandlungen der Preußischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse 1 (1929), reprinted in Gesammelte Abhandlungen I, 209– 266.