Past lectures

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 ∑𝑛≥0𝑛!𝑧𝑛.

References

[1] Y. André, Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, II. Transcendance sans transcendance, Ann. of Math. (2) 151 no. 2 (2000), 705–756.

[2] J. Fresán and P. Jossen, A non-​hypergeometric E-​function, Annals of Math. 194 (2021), 903–942.

[3] J. Fresán and P. Jossen, Exponential motives, preprint available at external 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.