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

E. Delaygue Université Claude Bernard Lyon 1, Villeurbanne, France
T. Rivoal CNRS and Université Grenoble Alpes, Grenoble, France
J. Roques CNRS and Université Grenoble Alpes, Grenoble, France
Available Formats:
Electronic ISBN: 978-1-4704-3635-3
Product Code: MEMO/246/1163.E
List Price: $75.00 MAA Member Price:$67.50
AMS Member Price: $45.00 Click above image for expanded view On Dwork’s$p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps E. Delaygue Université Claude Bernard Lyon 1, Villeurbanne, France T. Rivoal CNRS and Université Grenoble Alpes, Grenoble, France J. Roques CNRS and Université Grenoble Alpes, Grenoble, France Available Formats:  Electronic ISBN: 978-1-4704-3635-3 Product Code: MEMO/246/1163.E  List Price:$75.00 MAA Member Price: $67.50 AMS Member Price:$45.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2462016; 94 pp
MSC: 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.

• 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

• Requests

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Volume: 2462016; 94 pp
MSC: 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.

• 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
Review Copy – for reviewers who would like to review an AMS book
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