Niveau: Supérieur, Doctorat, Bac+8
ON THE INTEGRALITY OF THE TAYLOR COEFFICIENTS OF MIRROR MAPS C. KRATTENTHALER† AND T. RIVOAL 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 hyper- geometric 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?1q(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. 1. Introduction and statement of results 1.1. Mirror maps. Mirror maps have appeared quite recently in mathematics and phys- ics. Indeed, the term “mirror map” was coined in the late 1980s by physicists whose research in string theory led them to discover deep facts in algebraic geometry (e.g., given a Calabi–Yau threefold M , they constructed another Calabi–Yau threefold, the “mirror” of M , whose properties can be used to enumerate the rational curves on M).
- implies theorem
- integers
- calabi–yau complete
- h2m ?
- mirror map
- kn z
- corresponding mirror
- rather sharp integrality
- results