This material has been published in J. Austral. Math. Soc. 92 (2012), 195-235, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Cambridge University Press. This material may not be copied or reposted without explicit permission.

Christian Krattenthaler and Tanguy Rivoal

Analytic properties of mirror maps

(40 pages)

Abstract. We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [Nucl. Phys. B359 (1991), 21-74] associated to the quintic hypersurface in P4(C). In [Duke Math. J. 151 (2010), 175-218], we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of convergence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

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