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Analytic Continuation
Number Theory
Radius of Convergence
Sign Pattern
Taylor Expansion
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Analytic properties of mirror maps
Analytic properties of mirror maps,Christian Krattenthaler,Tanguy Rivoal
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Analytic properties of mirror maps
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Christian Krattenthaler
,
Tanguy Rivoal
We consider a multiparameter family of canonical coordinates and mirror maps o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604616]. This family includes many of the known onevariable 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), 2174] associated to the quintic hypersurface in $\mathbb P^4(\mathbb C)$. In [Duke Math. J. 151 (2010), 175218], we proved that all coeffi\ cients 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.
Published in 2011.
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