G-functions and applications

Gabriele Bogo
Friday, February 3, 2017 - 14:00

A G-function is an analytic function $\sum_{n=0}^{+\infty} c_n z^n$ which is a solution of a linear differential equation, and such that the coefficients $c_n$ have some "nice" (= nearly integrality) properties.These functions were introduced to study transcendence problems in number theory, but also have some geometric applications. Moreover, they are strongly related to linear differential operators of geometric type (Bombieri-Dwork conjecture).In the talk I will introduce all these objects and present two main applications: Apery’s proof of irrationality of $\zeta(3)$ and an alternative proof, due to D. and G. Chudnovsky, of Falting’s theorem on isogeny classes of elliptic curves. If time perimts, I will speak about uniformization of algebraic curves and G-functions.

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