On Dwork's P-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps (Memoirs of the American Mathematical Society)

Delaygue, E./ Rivoal, T./ Roques, J.

American Mathematical Society（2017/03発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 94 p.
• 言語 ENG
• 商品コード 9781470423001
• DDC分類 512.74

Full Description

Using Dwork's theory, the authors prove a broad generalization of his famous p-adic formal congruences theorem. This enables them to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters in particular, they hold for any prime number p and not only for almost all primes. Furthermore, using Christol's functions, the authors provide an explicit formula for the ``Eisenstein constant'' of any hypergeometric series with rational parameters.

As an application of these results, the authors obtain an arithmetic statement ``on average'' of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.

Contents

Introduction
Statements of the main results
Structure of the paper
Comments on the main results, comparison with previous results and open questions
The $p$-adic valuation of Pochhammer symbols
Proof of Theorem 4
Formal congruences
Proof of Theorem 6
Proof of Theorem 9
Proof of Theorem 12
Proof of Theorem 8
Proof of Theorem 10
Proof of Corollary 14
Bibliography