Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Standard conjecture of Hodge type for powers of abelian varieties

Thomas Agugliaro

TL;DR

The paper proves the standard conjecture of Hodge type ($SCHT$) for powers of abelian threefolds and, via a Frobenius-rank criterion, for powers of simple abelian varieties of prime dimension over finite fields. It develops a novel framework using tannakian categories, an archimedean real fiber functor to real isocrystals, and a classification of simple Lefschetz motives through enriched Frobenius eigenvalues, enabling a decomposition that reduces positivity to rank-2 exotic motives. A key contribution is connecting two realizations to compare quadratic forms and applying Ancona–Marmora’s positivity results to deduce $SCHT$ from archimedean data and motivic decompositions. The results yield infinitely many new examples of varieties whose powers satisfy $SCHT$ and provide explicit criteria involving Frobenius rank, CM data, and endomorphism centers, with potential implications for related standard conjectures and Tate theory.

Abstract

We prove that the standard conjecture of Hodge type holds for powers of abelian threefolds. Along the way, we also prove the conjecture for powers of simple abelian variety of prime dimension over finite fields, and in other related cases based on the notion of Frobenius rank of Lenstra-Zarhin. The main tool is a result comparing two real fiber functors on tannakian categories. A second tool is a new an explicit description of simple Lefschetz motives over finite fields, in terms of ''enriched'' Frobenius eigenvalues.

Standard conjecture of Hodge type for powers of abelian varieties

TL;DR

The paper proves the standard conjecture of Hodge type () for powers of abelian threefolds and, via a Frobenius-rank criterion, for powers of simple abelian varieties of prime dimension over finite fields. It develops a novel framework using tannakian categories, an archimedean real fiber functor to real isocrystals, and a classification of simple Lefschetz motives through enriched Frobenius eigenvalues, enabling a decomposition that reduces positivity to rank-2 exotic motives. A key contribution is connecting two realizations to compare quadratic forms and applying Ancona–Marmora’s positivity results to deduce from archimedean data and motivic decompositions. The results yield infinitely many new examples of varieties whose powers satisfy and provide explicit criteria involving Frobenius rank, CM data, and endomorphism centers, with potential implications for related standard conjectures and Tate theory.

Abstract

We prove that the standard conjecture of Hodge type holds for powers of abelian threefolds. Along the way, we also prove the conjecture for powers of simple abelian variety of prime dimension over finite fields, and in other related cases based on the notion of Frobenius rank of Lenstra-Zarhin. The main tool is a result comparing two real fiber functors on tannakian categories. A second tool is a new an explicit description of simple Lefschetz motives over finite fields, in terms of ''enriched'' Frobenius eigenvalues.
Paper Structure (21 sections, 36 theorems, 118 equations)

This paper contains 21 sections, 36 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Tools and idea of the proof of \ref{['thm:intro-frob-rank']}
  4. Organization
  5. Conventions
  6. Number fields and $p$-adic fields
  7. Motives
  8. Realization functors
  9. Lefschetz motives
  10. Comparison and specializations
  11. Algebraic cycles and motives
  12. Algebraic cycles and the standard conjecture of Hodge type
  13. Review on tannakian categories:
  14. Motives and the standard conjecture of Hodge type
  15. Enriched eigenvalues and Lefschetz motives
  16. ...and 6 more sections

Key Result

Theorem 1.2

The standard conjecture of Hodge type holds for powers of simple abelian varieties over $\overline{\mathbb{F}}_{p} \xspace$ of prime dimension.

Theorems & Definitions (109)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Hard Lefschetz
  • Definition 3.4
  • Proposition 3.5
  • ...and 99 more