On refined conjectures of Birch and Swinnerton-Dyer type for Hasse-Weil-Artin L-series / David Burns, Daniel Macias Castillo.

Format:

Book

Author/Creator:

Burns, David, 1963- author.

Macias Castillo, Daniel, author.

Series:

Memoirs of the American Mathematical Society ; v. 1482.

Memoirs of the American Mathematical Society, 0065-9266 ; Number 1482

Language:

English

Subjects (All):

Number theory.

Geometry, Algebraic.

Physical Description:

v, 156 pages ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2024.

Summary:

We consider refined conjectures of Birch and Swinnterdon-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult Archimedean considerations related to refinements of the conjecture of Deligne and Gross and delicate p-adic congruence relations that involve the bi-extension height pairing of Mazur and Tate and are related to key aspects of noncommutative Iwasawa theory. In important special cases we provide strong evidence, both theoretical and numerical, in support of the conjectures.

Contents:

Chapter 1. Introduction

Chapter 2. Selmer complexes

Chapter 3. The refined Birch and Swinnerton-Dyer conjecture

Chapter 4. Periods and Galois-Gauss sums

Chapter 5. Local points on ordinary varieties

Chapter 6. Classical Selmer complexes and refined BSD

Chapter 7. Euler characteristics and Galois structures

Chapter 8. Abelian congruence relations and module structures

Chapter 9. Abelian congruence relations and height pairings

Chapter 10. Height pairing comparisons

Chapter 11. Modular symbols

Chapter 12. Heegner points

Appendix A. Refined BSD and equivariant Tamagawa numbers

Appendix B. Poitou-Tate duality

AppendixC. Bockstein homomorphisms.

Notes:

"May 2024 ; volume 297 ; number 1482 (first of 5 numbers)."

Includes bibliographical references (pages 151-156).

ISBN:

1470469669

9781470469665

OCLC:

1437988548