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Tate classes on self-products of Abelian varieties over finite fields

Yuri G. Zarhin

Abstract

We deal with $g$-dimensional abelian varieties $X$ over finite fields. We prove that there is an universal constant (positive integer) $N=N(g)$ that depends only on $g$ that enjoys the following properties. If a certain self-product of $X$ carries an exotic Tate class then the self-product $X^{2N}$of $X$ also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.

Tate classes on self-products of Abelian varieties over finite fields

Abstract

We deal with -dimensional abelian varieties over finite fields. We prove that there is an universal constant (positive integer) that depends only on that enjoys the following properties. If a certain self-product of carries an exotic Tate class then the self-product of also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.

Paper Structure

This paper contains 7 sections, 24 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiplicative relations between Weil numbers
  4. Frames and Skeletons of Abelian Varieties over Finite Fields
  5. Applications of Gordan's Lemma
  6. Linear Algebra
  7. Tate forms

Key Result

Theorem 1.4

Let $g$ be a positive integer. There exists a positive integer $N=N(g)$ that depends only on $g$ and enjoys the following property. Let $X$ be a $g$-dimensional abelian variety over a finite field $k$ such that there exists a nontrivial admissible function $R_X \to {\mathbb Z}$. Then there exists a

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.4
  • Definition 1.5
  • Lemma 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Definition 2.1
  • ...and 56 more