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For the hypergeometric series with parameters from the rational fields, thereis an effective criterion due to Christol to decide whether the hypergeometricseries is N-integral or not. Christol criterion is a basic and vital tool inthe recent striking work of Delaygue, Rivoal and Roques on the N-integrality ofthe hypergeometric mirror maps with rational parameters. In this paper, wedevelop a systematic theory on the N-integrality of the hypergeometric serieswith parameters from quadratic fields. We first present a detailed $p$-adicanalysis to set up a criterion of the $p$-adic integrality of thehypergeometric series with parameters from rational fields. Consequently, wepresent two equivalent statements for the hypergeometric series with parametersfrom algebraic number fields to be N-integral. Finally, by using these results,introducing a new function that extends the Christol's function and developinga further $p$-adic analysis, we establish a criterion for the N-integrality ofthe hypergeometric series with parameters from the quadratic fields. In theprocess, there are two important ingredients. One is the uniform distributionresult of roots of a quadratic congruence which is due to Duke, Friedlander andIwaniec together with Toth. Another one is an upper bound on the number ofsolutions of polynomial congruences given by Stewart in 1991.

Criterion for the integrality of hypergeometric series with parameters from quadratic fields