Electronic ISBN:  9781470436353 
Product Code:  MEMO/246/1163.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 246; 2016; 94 ppMSC: Primary 11; Secondary 14; 33;
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. 
Table of Contents

Chapters

1. Introduction

2. Statements of the main results

3. Structure of the paper

4. Comments on the main results, comparison with previous results and open questions

5. The $p$adic valuation of Pochhammer symbols

6. Proof of Theorem

7. Formal congruences

8. Proof of Theorem

9. Proof of Theorem

10. Proof of Theorem

11. Proof of Theorem

12. Proof of Theorem

13. Proof of Corollary


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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.

Chapters

1. Introduction

2. Statements of the main results

3. Structure of the paper

4. Comments on the main results, comparison with previous results and open questions

5. The $p$adic valuation of Pochhammer symbols

6. Proof of Theorem

7. Formal congruences

8. Proof of Theorem

9. Proof of Theorem

10. Proof of Theorem

11. Proof of Theorem

12. Proof of Theorem

13. Proof of Corollary