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 $n\ge0$, 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.
Document type :
Journal articles
Liste complète des métadonnées

Cited literature [17 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01325350
Contributor : Tanguy Rivoal <>
Submitted on : Thursday, June 2, 2016 - 10:36:19 AM
Last modification on : Monday, March 4, 2019 - 2:55:10 PM
Document(s) archivé(s) le : Saturday, September 3, 2016 - 10:41:12 AM

Files

dengfndef.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01325350, version 1
  • ARXIV : 1606.00706

Citation

Stéphane Fischler, Tanguy Rivoal. On the denominators of the Taylor coefficients of G-functions. Kyushu Journal of Mathematics, Kyushu University, 2017, 71, 2, pp.287-298. ⟨hal-01325350⟩

Share

Metrics

Record views

215

Files downloads

49