GAAG seminar - On G-functions of differential order 2

G-functions are power series that solve a linear differential equation and satisfy some growth conditions of arithmetic nature. Those 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. (Joint work with Josh Lam and Yichen Qin)