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401-5731-DRL 2 Credits DR D-MATH

E-Functions and Geometry

Lecturers & Examiners: Prof. Dr. Javier Fresán
Doctoral students of I-Math (UZH) need to send an email to Jessica Bolsinger ( ) with the course number. The email should have the subject „Graduate course registration (ETH)“.
VVZ CR n/a

Last Updated: 2026-02-05 16:14:51

Abstract

Nachdiplom lecture

Objective

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.

Content

E-functions were introduced by Siegel in a landmark 1929 paper 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 which determines the structure of the differential equations they satisfy. More recently, the links with arithmetic geometry and especially the theory of exponential periods 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. Many mysteries remain, however.

General Information

Language
English
Levels
DR

Examination

Type
ungraded semester performance

Course Components

Type Title Time & Place Hours
lecture E-​Functions and Geometry
If you would like to attend the lecture please register by 22 September. For the registration form see
  • Thu 10:15-12:00 (HG G 43)
26 h semesterly

Offered In