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The fullness conjectures for products of elliptic curves

Bruno Kahn, with an appendix by Cyril Demarche

TL;DR

This work proves André's fullness-type conjectures (Hodge, Tate, de Rham–Betti, and Ogus) for products $X$ of elliptic curves, showing that all classes of interest are generated in degree $2$ and that the relevant motivic Galois groups satisfy $G_{\omega}(X)=G_H(X)$ in the strong sense. The method combines Chow-Lefschetz motives with a Tannakian formalism, a two-step approach (enrichment axioms plus a descent argument), and a descent mechanism to transfer results from a large ground field to the base field. A key technical feat is a two-pronged reduction: (i) verify the conjectures in codimension $1$ and for CM/ordinary elliptic factors, and (ii) descend to arbitrary fields via a general descent proposition. The results extend to abelian varieties isogenous to products of elliptic curves and illuminate the structure of cycle algebras and period relations, providing a unified, simpler proof in this special but important case and clarifying the roles of CM, ordinary, and non-CM factors in the motivic picture.

Abstract

We prove all conjectures from chapter 7 of Yves André's book on motives in the case of products of elliptic curves. The proofs given here are simpler and more uniform than the previous proofs in known cases.

The fullness conjectures for products of elliptic curves

TL;DR

This work proves André's fullness-type conjectures (Hodge, Tate, de Rham–Betti, and Ogus) for products of elliptic curves, showing that all classes of interest are generated in degree and that the relevant motivic Galois groups satisfy in the strong sense. The method combines Chow-Lefschetz motives with a Tannakian formalism, a two-step approach (enrichment axioms plus a descent argument), and a descent mechanism to transfer results from a large ground field to the base field. A key technical feat is a two-pronged reduction: (i) verify the conjectures in codimension and for CM/ordinary elliptic factors, and (ii) descend to arbitrary fields via a general descent proposition. The results extend to abelian varieties isogenous to products of elliptic curves and illuminate the structure of cycle algebras and period relations, providing a unified, simpler proof in this special but important case and clarifying the roles of CM, ordinary, and non-CM factors in the motivic picture.

Abstract

We prove all conjectures from chapter 7 of Yves André's book on motives in the case of products of elliptic curves. The proofs given here are simpler and more uniform than the previous proofs in known cases.
Paper Structure (12 sections, 18 theorems, 12 equations)

This paper contains 12 sections, 18 theorems, 12 equations.

Table of Contents

  1. Notation
  2. Review of the de Rham-Betti and the Ogus conjectures
  3. Background
  4. Chow-Lefschetz motives
  5. The case of a large ground field
  6. Descent
  7. Proof of Theorem \ref{['t0']}
  8. The Hodge conjecture andre
  9. The Tate conjecture andre
  10. The de Rham-Betti conjecture andre
  11. The Ogus conjecture
  12. Appendix: sections of algebraic gerbes

Key Result

Theorem 1

All the above-mentioned conjectures hold for $X$, in the strong sense that the cohomology classes coming from algebraic cycles are generated by those of degree $2$. In particular, the Tate conjecture holds for $X$ over any finitely generated field $k$.

Theorems & Definitions (38)

  • Theorem 1
  • Corollary 2: see Proposition \ref{['p3.1']}
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 28 more