ON THE INTEGRALITY OF THE TAYLOR COEFFICIENTS OF MIRROR MAPS II

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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.

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Wolstenholme's theorem

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 seriesq(z) =zexp(G(z)/F(z)), whereF(z) andG(z) + log(z)F(z) are speciﬁc solutions of certain hypergeometric diﬀerential equations with maximal unipotent monodromy atz More= 0.the question of ﬁnding the largest integer precisely, we address vsuch that the Taylor coeﬃcients of (z−1q(z))1/vare still integers. particular, we In determine the Dwork–Kontsevich sequence (uN)N≥1, whereuNis the largest integer such thatqN(z)1/uNis a series with integer coeﬃcients, whereqN(z) = exp(GN(z)/FN(z)), FN(z) =Pm∞=0(N m)!zm/m!NandGN(z) =P∞m=1(HN m−Hm)(N m)!zm/m!N, with Hndenoting then-th harmonic number, conditional on the conjecture that there are no prime numberpand integerNsuch that thep-adic valuation ofHN−1 is strictly greater than 3.

1.Introduction and statement of results

The present article is a sequel to our article [6], where we proved general results con-cerning the integrality properties of mirror maps. We shall prove here stronger integrality assertions for certain special cases that appear frequently in the literature. For any vector N= (N, . . . , N) (withkoccurrences ofN), whereNis a positive integer, let us deﬁne the power series FN(z) =∞X(mN!mkN)!kzm(1.1) m=0 and ∞ GN(z) =XkN(HN m−Hm()mN!mk)N!kzm,(1.2) m=1 withHn:=Pni=11idenoting then functions-th harmonic number. TheFN(z) andGN(z) + log(z)FN(zsame hypergeometric diﬀerential equation with maximal) are solutions of the unipotent monodromy atz basis of solutions with at most logarithmic singularities A= 0.

Date: October 14, 2009. 2000Mathematics Subject Classiﬁcation.Primary 11S80; Secondary 11J99 14J32 33C20. Key words and phrases.Calabi–Yau manifolds, integrality of mirror maps,p-adic analysis, Dwork’s theory, harmonic numbers, hypergeometric diﬀerential equations. †Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and grant S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. 1

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