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On the new theory of 1-motives

Cristian D. Gonzalez-Aviles

Abstract

This is a letter (not intended for publication in a regular journal) written in response to two referees of my preprint "Local duality theorems for commutative algebraic groups". In it, I discuss possible applications of the new theory of 1-motives introduced in that preprint. One of these is a new approach to the BSD conjecture for abelian varieties over global fields (developed here only partially) which builds on Bloch's well-known volume-theoretic interpretation of the conjecture in the number field case.

On the new theory of 1-motives

Abstract

This is a letter (not intended for publication in a regular journal) written in response to two referees of my preprint "Local duality theorems for commutative algebraic groups". In it, I discuss possible applications of the new theory of 1-motives introduced in that preprint. One of these is a new approach to the BSD conjecture for abelian varieties over global fields (developed here only partially) which builds on Bloch's well-known volume-theoretic interpretation of the conjecture in the number field case.
Paper Structure (9 sections, 2 theorems, 40 equations)

This paper contains 9 sections, 2 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Algebraic groups over global function fields.
  3. Groups of Lourdeaux type
  4. Abelianization techniques for a class of smooth and connected algebraic groups
  5. Affine groups
  6. Non-affine groups
  7. Quasi- reductive groups
  8. One- motives and the BSD conjecture
  9. Concluding remarks

Key Result

Theorem 2.1

(Sansuc-Chernousov) If $k$ is a number field and $G$ is a connected linear algebraic $k$-group, then the Tate-Shafarevich set ${\!\hbox{\fontencoding{OT2} Sh}}^{ 1}(G)$ is equipped with a canonical abelian group structure and there exists a canonical perfect pairing of finite abelian groups Consequently, if $X$ is a torsor under $G$ over $k$, then the Brauer-Manin obstruction attached to ${\!\hbo

Theorems & Definitions (11)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.4
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Remark 2.12
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 1 more