Hossein Movasati, Khosro Monsef Shokri
We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. We prove its consequence on the $N$-integrality of the mirror map for the particular cases $1\leq n\leq 4$. We also give a straightforward and short proof of the only if part of the conjecture for arbitrary $n$. This was a conjecture in mirror symmetry which was first proved in particular cases by Lian-Yau. The general format was formulated by Zudilin and finally established by Krattenthaler-Rivoal. For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle groups with a cusp, and for $n=4$ we prove that 14 examples of hypergeometric Calabi-Yau equations are the full classification of hypergeometric mirror maps with integral coefficients. For our purpose we state and prove a refinement of a theorem of Dwork which largely simplifies many existing proofs in the literature. As a by-product we get the integrality of the corresponding algebra of modular-type functions. These are natural generalizations of the algebra of classical modular and quasi-modular forms in the case $n=2$.
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http://arxiv.org/abs/1306.5662
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