Title: On G-functions of differential order 2

Abstract: (Joint work with Josh Lam and Yichen Qin) G-functions are power series that solve a linear differential equation and satisfy some growth conditions of arithmetic nature. They have close ties to algebraic geometry because of the result that period functions associated with pencils of algebraic varieties are complex linear combinations of G-functions. Those G-functions which are solution of a differential equation of order 1 are algebraic of a rather particular shape. A rich family of G-functions of differential order 2 is given by algebraic substitutions of the classical Gauss hypergeometric series. However, I will argue that G-functions of order 2 are infinitely much richer than those. Indeed, there exist infinitely many non-equivalent differential equations of order 2 whose solutions include a G-function that is not a polynomial in algebraic substitutions of hypergeometric series. This solves Siegel's problem for G-functions, as formulated by Fischler and Rivoal, and answers a question of Krammer in connection to his counterexample to a conjecture by Dwork.