ON THE DENOMINATORS OF THE TAYLOR COEFFICIENTS OF G-FUNCTIONS

Abstract

Let ∑^∞_n=0 a_n zⁿ ∈ Q[[z]] be a G-function, and, for any n ≥ 0, let δ_n ≥ 1 denote the least integer such that δ_na₀, δ_na₁,..., δ_na_n are all algebraic integers. By the definition of a G-function, there exists some constant c ≥ 1 such that δ_n ≤ cⁿ⁺¹ for all n ≥ 0. In practice, it is observed that δ_n always divides D^s_bn C ⁿ⁺¹ where D_n = lcm{1, 2,..., n}, b, C are positive integers and s ≥ 0 is an integer. We prove that this observation holds for any G-function provided the following conjecture is assumed: let K be a number field, and L ∈ K[z,d/dz] be a G-operator; then the generic radius of solvability R_v (L) is equal to one, for all finite places v of 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.