| 概要 |
Let Σ^^∞__〈n=0〉a_nZ^n∈Q^^-[[Z]] be a G-function, and, for any n ≥0, let δ_n≥1 denote the least integer such that δ_n, a_0, δ_n a_1, …, δ_na_n are all algebraic integers. By the definition of a G-funct...ion, there exists some constant c≥1 such that δ_n≤c〈n+1〉 for all n≥0. In practice, it is observed that δ_n always divides D^s_〈bn〉C^〈n+1〉 where D_n=1cm｛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, 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.続きを見る
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Mendeley出力
