**Memoirs of the American Mathematical Society**

2016;
94 pp;
Softcover

MSC: Primary 11;
Secondary 14; 33

Print ISBN: 978-1-4704-2300-1

Product Code: MEMO/246/1163

List Price: $75.00

AMS Member Price: $45.00

MAA member Price: $67.50

**Electronic ISBN: 978-1-4704-3635-3
Product Code: MEMO/246/1163.E**

List Price: $75.00

AMS Member Price: $45.00

MAA member Price: $67.50

# On Dwork’s \(p\)-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps

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*E. Delaygue; T. Rivoal; J. Roques*

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

# Table of Contents

## On Dwork's $p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Statements of the main results 714
- Chapter 3. Structure of the paper 1320
- Chapter 4. Comments on the main results, comparison with previous results and open questions 1522
- Chapter 5. The 𝑝-adic valuation of Pochhammer symbols 2532
- Chapter 6. Proof of Theorem 4 3744
- Chapter 7. Formal congruences 3946
- Chapter 8. Proof of Theorem 6 4754
- Chapter 9. Proof of Theorem 9 7380
- Chapter 10. Proof of Theorem 12 7784
- Chapter 11. Proof of Theorem 8 7986
- Chapter 12. Proof of Theorem 10 8996
- Chapter 13. Proof of Corollary 14 9198
- Bibliography 93100
- Back Cover Back Cover1106