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Galois groups of simple abelian varieties over finite fields and exceptional Tate classes

Santiago Arango-Piñeros, Sam Frengley, Sameera Vemulapalli

TL;DR

This work develops a concrete, combinatorial framework to study the Tate conjecture for abelian varieties over finite fields by encoding the Galois action on Frobenius eigenvalues into weighted permutation representations. It provides a practical criterion to detect exceptional Tate classes, connects angle ranks to these representations, and proves Tate conjecture cases beyond maximal angle rank, including ordinary geometrically simple prime-dimension varieties with commutative endomorphism algebras. A central achievement is an algorithmic construction of $q$-Weil numbers realizing prescribed CM-field endomorphism centers, with thorough analysis of when these realizations yield geometrically simple abelian varieties; this ties into strong inverse Galois problems and, under tame-Grunwald-type conjectures, yields broad conditional realizations. The paper also offers explicit classifications for small dimensions ($g\le 6$), including concrete examples and a detailed exploration of CM-field realizations, thereby extending prior results of Tankeev, Zarhin, Lenstra–Zarhin, and Dupuy–Kedlaya–Zureick-Brown. Overall, the results not only advance Tate conjecture cases but also illuminate constructive pathways to endomorphism-algebra centers via CM fields and provide a computational toolkit for identifying exceptional Tate classes.

Abstract

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate conjecture in cases when the angle rank is non-maximal. Our primary tool is a precise combinatorial condition which, given a geometrically simple abelian variety $A/\mathbf{F}_q$ with commutative endomorphism algebra, describes whether $A$ has exceptional classes (i.e., $\mathrm{Gal}( \bar{\mathbf{F}}_q/\mathbf{F}_q)$-invariant classes in $H_{\text{ét}}^{2r}(A_{\bar{\mathbf{F}}_q}, \mathbf{Q}_\ell(r))$ not contained in the span of classes of intersections of divisors). The criterion depends only on the Galois group of the minimal polynomial of Frobenius and its action on the Newton polygon of $A$. Our tools provide substantial control over the isogeny invariants of $A$, allowing us to prove a number of new results. Firstly, we provide an algorithm which, given a Newton polygon and CM field, determines if they arise from a geometrically simple abelian variety $A/\mathbf{F}_q$ and, if so, outputs one such $A$. As a consequence we show that every CM field occurs as the center of the endomorphism algebra of an abelian variety $A/\mathbf{F}_q$. Secondly, we refine a result of Tankeev and Dupuy--Kedlaya--Zureick-Brown on angle ranks of abelian varieties. In particular, we show that ordinary geometrically simple varieties of prime dimension have maximal angle rank.

Galois groups of simple abelian varieties over finite fields and exceptional Tate classes

TL;DR

This work develops a concrete, combinatorial framework to study the Tate conjecture for abelian varieties over finite fields by encoding the Galois action on Frobenius eigenvalues into weighted permutation representations. It provides a practical criterion to detect exceptional Tate classes, connects angle ranks to these representations, and proves Tate conjecture cases beyond maximal angle rank, including ordinary geometrically simple prime-dimension varieties with commutative endomorphism algebras. A central achievement is an algorithmic construction of -Weil numbers realizing prescribed CM-field endomorphism centers, with thorough analysis of when these realizations yield geometrically simple abelian varieties; this ties into strong inverse Galois problems and, under tame-Grunwald-type conjectures, yields broad conditional realizations. The paper also offers explicit classifications for small dimensions (), including concrete examples and a detailed exploration of CM-field realizations, thereby extending prior results of Tankeev, Zarhin, Lenstra–Zarhin, and Dupuy–Kedlaya–Zureick-Brown. Overall, the results not only advance Tate conjecture cases but also illuminate constructive pathways to endomorphism-algebra centers via CM fields and provide a computational toolkit for identifying exceptional Tate classes.

Abstract

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate conjecture in cases when the angle rank is non-maximal. Our primary tool is a precise combinatorial condition which, given a geometrically simple abelian variety with commutative endomorphism algebra, describes whether has exceptional classes (i.e., -invariant classes in not contained in the span of classes of intersections of divisors). The criterion depends only on the Galois group of the minimal polynomial of Frobenius and its action on the Newton polygon of . Our tools provide substantial control over the isogeny invariants of , allowing us to prove a number of new results. Firstly, we provide an algorithm which, given a Newton polygon and CM field, determines if they arise from a geometrically simple abelian variety and, if so, outputs one such . As a consequence we show that every CM field occurs as the center of the endomorphism algebra of an abelian variety . Secondly, we refine a result of Tankeev and Dupuy--Kedlaya--Zureick-Brown on angle ranks of abelian varieties. In particular, we show that ordinary geometrically simple varieties of prime dimension have maximal angle rank.
Paper Structure (24 sections, 33 theorems, 67 equations, 12 tables, 1 algorithm)

This paper contains 24 sections, 33 theorems, 67 equations, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Tate conjecture and Galois groups
  3. Angle ranks of abelian varieties
  4. Reconstruction theorems
  5. Outline of the paper
  6. Notation and conventions
  7. Acknowledgments
  8. The Galois action on q-Weil numbers
  9. The Galois action on the Frobenius eigenvalues
  10. A necessary condition for $\rho$ to arise from an abelian variety
  11. A group-theoretic formula for div_A
  12. Angle ranks of geometrically simple abelian varieties
  13. An explicit criterion for the existence of exceptional classes
  14. Applications of Theorem 4.4
  15. The proof of ??
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $A/\mathbf{F}_q$ be a geometrically simple abelian variety of dimension $g$ with commutative endomorphism algebra. Then, for each $1 \leq r \leq g$, the algebra $H_{\textnormal{\'et}}^{2r}(A_{\mkern 1.5mu\overline{\mkern-1.5mu\mathbf{F}\mkern-1.5mu}\mkern 1.5mu_q}, \mathbf{Q}_\ell(r))^{G_{\mathb

Theorems & Definitions (78)

  • Conjecture 1.1: The Tate conjecture for $A$
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6: Refined Tankeev's Theorem
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • Definition 2.1
  • ...and 68 more