G-functions and Holonomy Bounds

A G-function, in the sense of Siegel, is a power series with rational coefficients that on the one hand has nice arithmetic properties (the LCM of the denominators of the first \(n\) coefficients grows at most exponentially) and on the other hand has nice analytic properties (it is the solution of a linear ODE with coefficients in \(ℚ(x)\)). These functions are expected to have rich connections with geometry.

In recent years, Dimitrov, Tang, and the speaker have been exploring precisely the tension between these analytic and arithmetic properties. In certain prescribed contexts, we are able to prove a ``holonomy bound'' showing that power series with certain properties must be G-functions of small order.

The goal of this talk is to explore the limits of our results, to raise a number of basic open questions about G-functions, and also, perhaps, to discuss some unexpected applications of our results, for example to the effective \(S\)-unit equation or to a new proof of the transcendence of \(π\).