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On Dwork’s

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2017Copyright date: ©2016Edition: 1st edDescription: 1 online resource (106 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470436353
Subject(s): Genre/Form: Additional physical formats: Print version:: On Dwork’s DDC classification:
  • 512.74
LOC classification:
  • QA564.D453 2017
Online resources:
Contents:
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Statements of the main results -- 2.1. -integrality of _{ , } -- 2.2. Generalizations of Dwork's congruence -- 2.3. -Integrality of _{ , } -- 2.4. -Integrality of _{ , } -- Chapter 3. Structure of the paper -- Chapter 4. Comments on the main results, comparison with previous results and open questions -- 4.1. Comments on Theorem 3, Theorem 4 and on the hypothesis _{ , } -- 4.2. Comparison with previous results -- 4.3. Open questions -- 4.4. A corrected version of a lemma of Lang -- Chapter 5. The -adic valuation of Pochhammer symbols -- 5.1. Christol's criterion for the -integrality of _{ , } -- 5.2. Dwork's map _{ } -- 5.3. Analogues of Landau functions -- Chapter 6. Proof of Theorem 4 -- Chapter 7. Formal congruences -- 7.1. Proof of Theorem 30 -- Chapter 8. Proof of Theorem 6 -- 8.1. Algebras of functions taking values into ℤ_{ } -- 8.2. Proof of Theorem 6 -- Chapter 9. Proof of Theorem 9 -- Chapter 10. Proof of Theorem 12 -- 10.1. A -adic reformulation of Theorem 12 -- Chapter 11. Proof of Theorem 8 -- 11.1. Proof of Assertion ( ) of Theorem 8 -- 11.2. Proof of Assertion ( ) of Theorem 8 -- 11.3. Last step in the proof of Theorem 8 -- Chapter 12. Proof of Theorem 10 -- Chapter 13. Proof of Corollary 14 -- Bibliography -- Back Cover.
Summary: 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.
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Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Statements of the main results -- 2.1. -integrality of _{ , } -- 2.2. Generalizations of Dwork's congruence -- 2.3. -Integrality of _{ , } -- 2.4. -Integrality of _{ , } -- Chapter 3. Structure of the paper -- Chapter 4. Comments on the main results, comparison with previous results and open questions -- 4.1. Comments on Theorem 3, Theorem 4 and on the hypothesis _{ , } -- 4.2. Comparison with previous results -- 4.3. Open questions -- 4.4. A corrected version of a lemma of Lang -- Chapter 5. The -adic valuation of Pochhammer symbols -- 5.1. Christol's criterion for the -integrality of _{ , } -- 5.2. Dwork's map _{ } -- 5.3. Analogues of Landau functions -- Chapter 6. Proof of Theorem 4 -- Chapter 7. Formal congruences -- 7.1. Proof of Theorem 30 -- Chapter 8. Proof of Theorem 6 -- 8.1. Algebras of functions taking values into ℤ_{ } -- 8.2. Proof of Theorem 6 -- Chapter 9. Proof of Theorem 9 -- Chapter 10. Proof of Theorem 12 -- 10.1. A -adic reformulation of Theorem 12 -- Chapter 11. Proof of Theorem 8 -- 11.1. Proof of Assertion ( ) of Theorem 8 -- 11.2. Proof of Assertion ( ) of Theorem 8 -- 11.3. Last step in the proof of Theorem 8 -- Chapter 12. Proof of Theorem 10 -- Chapter 13. Proof of Corollary 14 -- Bibliography -- Back Cover.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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