Javier Fresán: E-functions and Geometry (NTWS 204)
Channel: Number Theory Web Seminar
Date: 2024-03-25

Abstract: E-functions are power series which solve a differential equation and whose coefficients are algebraic numbers that satisfy certain growth conditions of arithmetic nature. They were introduced in Siegel's 1929 memoir on the applications of diophantine approximation with the goal of generalising the Hermite--Lindemann--Weierstrass theorem about the transcendence of the values of the exponential function at algebraic arguments. Besides the exponential, standard examples include the Bessel function and confluent hypergeometric series. After briefly surveying on the history of E-functions, I will present a joint work in progress with Peter Jossen where we prove that exponential period functions provide us with a rich geometric source of E-functions. The easiest examples, attached to polynomials of degree 4, already allowed us a couple of years ago to exhibit some E-functions which are not polynomial expressions in hypergeometric series, thus solving one of the problems in Siegel's original paper. 

Link to slides: https://drive.google.com/file/d/16pmNCInxKaEj8D5f2uX1QumzJCO6wG3c

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, March 21, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 22, 2024 (12am CST, 3am AEDT, 5am NZDT)