On the denominators of the Taylor coefficients of G-functions

TL;DR

This paper investigates the denominators of Taylor coefficients of G-functions, proposing a conjecture related to p-adic differential equations that, if true, explains observed divisibility patterns and links to the Bombieri-Dwork Conjecture.

Contribution

It establishes a conditional result on the divisibility of denominators of G-function coefficients, connecting it to a conjecture on G-operators and p-adic solvability.

Findings

Divisibility pattern holds under the conjecture for all G-functions.

Unconditional results are obtained for geometric differential operators.

The work links G-functions to p-adic differential equations and the Bombieri-Dwork Conjecture.

Abstract

Let $\sum\_{n=0}^\infty a\_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $\delta\_n\ge 1$ denote the least integer such that $\delta\_n a\_0, \delta\_n a\_1, ..., \delta\_n a\_n$ 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 $L\in \mathbb{K}[z,\frac{d }{d z}]$ 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.