On the denominators of the Taylor coefficients of G-functions

Abstract

Let $\sum \_{n=0}^{\infty }a\_n{z}^n\in \overline{\mathbb{Q}}[[z]]$ be a $G$-function, and, for any $n\ge 0$, let $\delta \_n\ge 1$ denote the least integer such that $\delta \_na\_0,\delta \_na\_1,...,\delta \_na\_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ such that $\delta \_n\le c^{n+1}$ for all $n\ge 0$. In practice, it is observed that $\delta \_n$ always divides $D\_{bn}^s{C}^{n+1}$ where $D\_n=lcm\{1,2,...,n\}$, $b,C$ are positive integers and $s\ge 0$ is an integer. We prove that this observation holds for any $G$-function provided the following conjecture is assumed: {\em Let $\mathbb{K}$ be a number field, and $L\in \mathbb{K}[z,\frac{d}{dz}]$ be a $G$-operator; then the generic radius of solvability $R\_v(L)$ is equal to 1, for all finite places $v$ of $\mathbb{K}$ except a finite number.} The proof makes use of very precise estimates in the theory of $p$-adic differential equations, in particular the Christol-Dwork Theorem. Our result becomes unconditional when $L$ is a geometric differential operator, a special type of $G$-operators for which the conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts that any $G$-operator is of geometric type, hence it implies the above conjecture.

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• Computing periods of rational integrals
• A fast algorithm for computing the characteristic polynomial of the p-curvature