No abstract

We show that the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)) are integers, where F(z) and G(z) + log(z)F(z) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z = 0. We also address the question of finding the largest integer u such that the Taylor coefficients of (z −1 q(z)) 1/u are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau and by Zudilin. In particular, we prove the general "integrality" conjecture of Zudilin about these mirror maps.

Let f be a G-function (in the sense of Siegel), and α be an algebraic number; assume that the value f (α) is a real number. As a special case of a more general result, we show that this number can be written as g (1), where g is a G-function with rational coefficients and arbitrarily large radius of convergence. As an application, we prove that quotients of such values are exactly the numbers which can be written as limits of sequences a n /b n , where ∞ n=0 a n z n and ∞ n=0 b n z n are G-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for ζ(3), and gives answers to questions asked by T. Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365].