2007/09/10 by Christian Krattenthaler, Krattenthaler, Christian, Tanguy Rivoal +1 · 2 citations
Mathematics · Physics and Astronomy · #11J99 #11S80 (Primary) #14J32 #33C20 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #hep-th #math.AG #math.NT #msc:11J99 #msc:11S80 #msc:14J32 #msc:33C20
paper · pdf · doi:10.48550/arxiv.0709.1432
AmS-LaTeX; 54 pages. This paper was cut in two separate papers, arXiv:0907.2577 and arXiv:0907.2578, and is thereby superseded by these two
openalex publication_date 2007/09/10 · arxiv created 2009/07/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the Taylor coefficients of the series \bf q(z)=zexp(\bf G(z)/\bf F(z)) are integers, where \bf F(z) and \bf G(z)+log(z) \bf 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\bf 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. A further outcome of the present study is the determination of the Dwork-Kontsevich sequence (uN)N≥1, where uN is the largest integer such that q(z)1/uN is a series with integer coefficients, where q(z)=exp(F(z)/G(z)), F(z)=∑m=0 ∞ (Nm)! zm/m!N and G(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.