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Abelian motives and Shimura varieties in nonzero characteristic

James S. Milne

Abstract

Much of the work on Shimura varieties over the last thirty years has been devoted to constructing the theory that would follow from a good notion of motives, one incorporating the Hodge, Tate, and standard conjectures. These conjectures are believed to be beyond reach, and may not even be correct as stated. I argue in this article that there exists a theory of motives, accessible to proof, weaker than Grothendieck's, but with many of the same consequences.

Abelian motives and Shimura varieties in nonzero characteristic

Abstract

Much of the work on Shimura varieties over the last thirty years has been devoted to constructing the theory that would follow from a good notion of motives, one incorporating the Hodge, Tate, and standard conjectures. These conjectures are believed to be beyond reach, and may not even be correct as stated. I argue in this article that there exists a theory of motives, accessible to proof, weaker than Grothendieck's, but with many of the same consequences.

Paper Structure

This paper contains 88 sections, 65 theorems, 177 equations.

Table of Contents

  1. Statement of the conjectures
  2. Equivalence of the conjectures
  3. Proof of the uniqueness of rational Tate classes
  4. Consequences of the conjectures
  5. Positive results
  6. A variational approach to proving Conjecture A
  7. Outline of the article
  8. Notation
  9. Preliminaries
  10. Tannakian categories
  11. CM abelian varieties
  12. Abelian motives
  13. Definition
  14. CM motives
  15. Abelian motives over the complex numbers
  16. ...and 73 more sections

Key Result

Theorem 1

Let $(\mathcal{R}^{\ast}(A))_{A\in\mathscr{s}{}}$ be a family of rational Tate classes, as in Conjecture C. Then, for all $A\in\mathscr{s}$, the map has image $\mathcal{T}^{\ast}(A)$, i.e., $\mathcal{R}{}^{\ast}(A)$ is a $\mathbb{Q}$-structure on $\mathcal{T}{}^{\ast}(A)$.

Theorems & Definitions (141)

  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2
  • Theorem 4
  • Proof 3
  • Theorem 1.7
  • Proof 4
  • Theorem 1.8
  • Proof 5
  • ...and 131 more