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Arithmetic properties of mirror map and quantum coupling. (English) Zbl 0867.14017
The mirror map and the quantum coupling are known to have many interesting number theoretic properties. For example, the Fourier coefficients of the mirror map are integral in all known cases. In this paper, the authors prove that some K3 mirror maps are algebraic over \(\mathbb{Q} (J)\) by solving the Schwarzian differential equation in terms of the \(J\)-function, which leads to a uniform proof that these mirror maps have integral Fourier coefficients. The authors also give another proof that those mirror maps are integral by using the fact that such maps are genus zero functions as Riemann mappings, and conjecture a connection between K3 mirror maps and the Thompson series. Finally, they discuss \(\text{mod } p\) congruences for the cases of threefolds and the quintics.

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
14M30 Supervarieties
14J30 \(3\)-folds
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