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Explicit Arithmetic of Jacobians of Generalized Legendre Curves over Global Function Fields.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2020Copyright date: ©2020Edition: 1st edDescription: 1 online resource (144 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470462536
Subject(s): Genre/Form: Additional physical formats: Print version:: Explicit Arithmetic of Jacobians of Generalized Legendre Curves over Global Function FieldsDDC classification:
  • 516.35199999999998
LOC classification:
  • QA565 .B474 2020
Online resources:
Contents:
Cover -- Title page -- Introduction -- Historical background -- The main results -- Overview of the paper -- Guide -- Notation -- Chapter 1. The curve, explicit divisors, and relations -- 1.1. A generalization of the Legendre curve -- 1.2. Explicit points and the visible subgroup -- 1.3. Relations -- 1.4. Torsion -- 1.5. First main theorem -- 1.6. Complement: Other curves -- Chapter 2. Descent calculations -- 2.1. The isogeny -- 2.2. The homomorphism \XminusT -- 2.3. The image of \XminusT -- 2.4. Proof of the main theorem -- Chapter 3. Minimal regular model, local invariants, and domination by a product of curves -- 3.1. Models -- 3.2. Local invariants of the Néron model -- 3.3. Domination by a product of curves -- Chapter 4. Heights and the visible subgroup -- 4.1. Height pairing -- 4.2. A group-theoretic pairing -- 4.3. Structure of the visible subgroup -- 4.4. Discriminants -- Chapter 5. The -function and the BSD conjecture -- 5.1. The -function -- 5.2. The conjecture of Birch and Swinnerton-Dyer for -- 5.3. Elementary calculation of the -function -- 5.4. Ranks -- Chapter 6. Analysis of [ ] and ( _{ })_{ ℴ } -- 6.1. Kodaira-Spencer and -torsion -- 6.2. Néron-Severi of \XX_{ } is torsion-free -- Chapter 7. Index of the visible subgroup and the Tate-Shafarevich group -- 7.1. Visible versus Mordell-Weil -- 7.2. Tamagawa number -- 7.3. Application of the BSD formula -- Chapter 8. Monodromy of ℓ-torsion and decomposition of the Jacobian -- 8.1. Statement of results -- 8.2. New and old -- 8.3. Endomorphism rings -- 8.4. The Λ-module structure of [ℓ] -- 8.5. Monodromy of [ ] -- 8.6. Independence -- 8.7. Conclusion -- Appendix A. An additional hyperelliptic family -- A.1. Introduction -- A.2. The BSD conjecture -- A.3. Descent -- A.4. Degree of the -function -- A.5. Additional remarks -- Bibliography -- Back Cover.
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Cover -- Title page -- Introduction -- Historical background -- The main results -- Overview of the paper -- Guide -- Notation -- Chapter 1. The curve, explicit divisors, and relations -- 1.1. A generalization of the Legendre curve -- 1.2. Explicit points and the visible subgroup -- 1.3. Relations -- 1.4. Torsion -- 1.5. First main theorem -- 1.6. Complement: Other curves -- Chapter 2. Descent calculations -- 2.1. The isogeny -- 2.2. The homomorphism \XminusT -- 2.3. The image of \XminusT -- 2.4. Proof of the main theorem -- Chapter 3. Minimal regular model, local invariants, and domination by a product of curves -- 3.1. Models -- 3.2. Local invariants of the Néron model -- 3.3. Domination by a product of curves -- Chapter 4. Heights and the visible subgroup -- 4.1. Height pairing -- 4.2. A group-theoretic pairing -- 4.3. Structure of the visible subgroup -- 4.4. Discriminants -- Chapter 5. The -function and the BSD conjecture -- 5.1. The -function -- 5.2. The conjecture of Birch and Swinnerton-Dyer for -- 5.3. Elementary calculation of the -function -- 5.4. Ranks -- Chapter 6. Analysis of [ ] and ( _{ })_{ ℴ } -- 6.1. Kodaira-Spencer and -torsion -- 6.2. Néron-Severi of \XX_{ } is torsion-free -- Chapter 7. Index of the visible subgroup and the Tate-Shafarevich group -- 7.1. Visible versus Mordell-Weil -- 7.2. Tamagawa number -- 7.3. Application of the BSD formula -- Chapter 8. Monodromy of ℓ-torsion and decomposition of the Jacobian -- 8.1. Statement of results -- 8.2. New and old -- 8.3. Endomorphism rings -- 8.4. The Λ-module structure of [ℓ] -- 8.5. Monodromy of [ ] -- 8.6. Independence -- 8.7. Conclusion -- Appendix A. An additional hyperelliptic family -- A.1. Introduction -- A.2. The BSD conjecture -- A.3. Descent -- A.4. Degree of the -function -- A.5. Additional remarks -- Bibliography -- Back Cover.

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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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