ON THE INTEGRALITY OF THE TAYLOR COEFFICIENTS OF MIRROR MAPS II

ON THE INTEGRALITY OF THE TAYLOR COEFFICIENTS OF MIRROR MAPS, II C. KRATTENTHALER† AND T. RIVOAL Abstract. We continue our study begun in “On the integrality of the Taylor coefficients of mirror maps” [Duke Math. J. (to appear)] of the fine integrality properties of the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)), 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. More precisely, we address the question of finding the largest integer v such that the Taylor coefficients of (z?1q(z))1/v are still integers. In particular, we determine the Dwork–Kontsevich sequence (uN )N≥1, where uN is the largest integer such that qN (z)1/uN is a series with integer coefficients, where qN (z) = exp(GN (z)/FN (z)), FN (z) = ∑∞ m=0(Nm)! zm/m!N and GN (z) = ∑∞ m=1(HNm ? Hm)(Nm)! zm/m!N , with Hn denoting the n-th harmonic number, conditional on the conjecture that there are no prime number p and integer N such that the p-adic valuation of HN ? 1 is strictly greater than 3.

• largest integer


• parameters can


• then


• kkn ?


• let vn


• integrality properties


• mirror map


• wolstenholme prime