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On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic

David Burns, Mahesh Kakde, Wansu Kim

TL;DR

This work develops a refined Birch and Swinnerton-Dyer framework for abelian varieties over global function fields by integrating leading-term interpolation of Hasse–Weil Artin L-series with a Galois-structure analysis of Selmer complexes within relative $K$-theory. The authors construct a canonical leading-term element in $K_1(b{R}[G])$ and formulate a precise boundary-map relation linking this term to arithmetic and coherent Euler characteristics, mirroring the equivariant Tamagawa-number philosophy. They prove substantial cases unconditionally (notably when a Tate–Shafarevich group is finite for some prime, and under semistability with tame ramification), and provide a robust framework that unifies syntomic/crystalline cohomology, Selmer complexes, and height pairings to derive both order-of-vanishing results and leading-term relations. The approach yields both new instances of refined BSD in ramified settings and a methodological bridge to ETNC-style conjectures for function fields, with explicit $K$-theoretic and cohomological machinery that can be further leveraged for generically ordinary abelian varieties. Overall, the paper advances a deep arithmetic refinement in positive characteristic by embedding L-term congruences, Selmer-structure restrictions, and coherent- versus arithmetic-data balance into a cohesive non-commutative $K$-theoretic conjecture and its verifications.

Abstract

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture for abelian varieties over global function fields. This refinement incorporates both families of congruences between the leading terms of Artin-Hasse-Weil $L$-series and also strong restrictions on the Galois structure of natural Selmer complexes and constitutes a precise analogue for abelian varieties over function fields of the equivariant Tamagawa number conjecture for abelian varieties over number fields. We then provide strong supporting evidence for this conjecture including giving a full proof, modulo only the assumed finiteness of Tate-Shafarevich groups, in an important class of examples.

On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic

TL;DR

This work develops a refined Birch and Swinnerton-Dyer framework for abelian varieties over global function fields by integrating leading-term interpolation of Hasse–Weil Artin L-series with a Galois-structure analysis of Selmer complexes within relative -theory. The authors construct a canonical leading-term element in and formulate a precise boundary-map relation linking this term to arithmetic and coherent Euler characteristics, mirroring the equivariant Tamagawa-number philosophy. They prove substantial cases unconditionally (notably when a Tate–Shafarevich group is finite for some prime, and under semistability with tame ramification), and provide a robust framework that unifies syntomic/crystalline cohomology, Selmer complexes, and height pairings to derive both order-of-vanishing results and leading-term relations. The approach yields both new instances of refined BSD in ramified settings and a methodological bridge to ETNC-style conjectures for function fields, with explicit -theoretic and cohomological machinery that can be further leveraged for generically ordinary abelian varieties. Overall, the paper advances a deep arithmetic refinement in positive characteristic by embedding L-term congruences, Selmer-structure restrictions, and coherent- versus arithmetic-data balance into a cohesive non-commutative -theoretic conjecture and its verifications.

Abstract

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture for abelian varieties over global function fields. This refinement incorporates both families of congruences between the leading terms of Artin-Hasse-Weil -series and also strong restrictions on the Galois structure of natural Selmer complexes and constitutes a precise analogue for abelian varieties over function fields of the equivariant Tamagawa number conjecture for abelian varieties over number fields. We then provide strong supporting evidence for this conjecture including giving a full proof, modulo only the assumed finiteness of Tate-Shafarevich groups, in an important class of examples.

Paper Structure

This paper contains 39 sections, 38 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Leading terms of Hasse-Weil-Artin $L$-series
  3. Arithmetic complexes
  4. Selmer groups
  5. Arithmetic cohomology
  6. Pro-$p$ subgroups
  7. Selmer complexes
  8. Coherent cohomology
  9. Statements of the conjecture and main results
  10. Relative $K$-theory
  11. The refined Birch and Swinnerton-Dyer Conjecture
  12. The main results
  13. Preliminary results
  14. A result in $K$-theory
  15. Consistency checks
  16. ...and 24 more sections

Key Result

Theorem 2.1

There exists a unique element $L^*_{U}(A_{L/K},1)$ of $K_1(\mathbb{R}[G])$ with the property that ${\rm Nrd}_{\mathbb{R}[G]}(L^*_{U}(A_{L/K}, 1))_\chi = L^*_U(A,\chi, 1)$ for all $\chi$ in ${\rm Ir}(G)$.

Theorems & Definitions (98)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 88 more