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Nondegeneracy and Sato-Tate Distributions of Two Families of Jacobian Varieties

Melissa Emory, Heidi Goodson

TL;DR

The paper advances the Sato–Tate program for two infinite families of Jacobians, proving nondegeneracy via Hodge theory and explicitly computing their Sato–Tate groups through twisted Lefschetz groups. By decomposing $ ext{Jac}(C_{2^m})$ into factors $ ext{Jac}(C'_{2^d+1})$ and analyzing CM endomorphism fields $F_d$, it derives precise identity components and component generators for each family, including explicit matrices that realize the component actions. It also provides moment statistics from the resulting Sato–Tate groups and validates these against numerical moments from normalized $L$-polynomials, enabling a practical check of equidistribution in the generalized Sato–Tate setting. The work yields new examples of stably nondegenerate Jacobians with noncyclic endomorphism fields and demonstrates how the factorization structure encodes the full Sato–Tate behavior, contributing to both the theory and computational verification of higher-dimensional Sato–Tate phenomena.

Abstract

We consider the curves $y^2=x^{2^m} -1$ and $y^2=x^{2^{d}+1}-x$ over the rationals. These curves are related via their associated Jacobian varieties in that the Jacobians of the latter appear as factors of the Jacobians of the former. One of the principal aims of this paper is to fully describe their Sato-Tate groups and distributions by determining generators of the component groups. In order to do this, we first prove the nondegeneracy of the two families of Jacobian varieties via their Hodge groups. We then use results relating Sato-Tate groups and twisted Lefschetz groups of nondegenerate abelian varieties to determine the generators of the associated Sato-Tate groups. The results of this paper add new examples to the literature of families of nondegenerate Jacobian varieties and of noncyclic component groups of Sato-Tate groups. Furthermore, we compute moment statistics associated to the Sato-Tate groups which can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized $L$-polynomials of the curves.

Nondegeneracy and Sato-Tate Distributions of Two Families of Jacobian Varieties

TL;DR

The paper advances the Sato–Tate program for two infinite families of Jacobians, proving nondegeneracy via Hodge theory and explicitly computing their Sato–Tate groups through twisted Lefschetz groups. By decomposing into factors and analyzing CM endomorphism fields , it derives precise identity components and component generators for each family, including explicit matrices that realize the component actions. It also provides moment statistics from the resulting Sato–Tate groups and validates these against numerical moments from normalized -polynomials, enabling a practical check of equidistribution in the generalized Sato–Tate setting. The work yields new examples of stably nondegenerate Jacobians with noncyclic endomorphism fields and demonstrates how the factorization structure encodes the full Sato–Tate behavior, contributing to both the theory and computational verification of higher-dimensional Sato–Tate phenomena.

Abstract

We consider the curves and over the rationals. These curves are related via their associated Jacobian varieties in that the Jacobians of the latter appear as factors of the Jacobians of the former. One of the principal aims of this paper is to fully describe their Sato-Tate groups and distributions by determining generators of the component groups. In order to do this, we first prove the nondegeneracy of the two families of Jacobian varieties via their Hodge groups. We then use results relating Sato-Tate groups and twisted Lefschetz groups of nondegenerate abelian varieties to determine the generators of the associated Sato-Tate groups. The results of this paper add new examples to the literature of families of nondegenerate Jacobian varieties and of noncyclic component groups of Sato-Tate groups. Furthermore, we compute moment statistics associated to the Sato-Tate groups which can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized -polynomials of the curves.
Paper Structure (21 sections, 15 theorems, 67 equations, 4 tables)

This paper contains 21 sections, 15 theorems, 67 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. An $\ell$-adic Construction of the Sato-Tate Group
  4. Hodge and Mumford Tate Groups
  5. Nondegenerate Abelian Varieties
  6. Nondegeneracy
  7. Decompositions of the Jacobians
  8. Nondegeneracy Results
  9. Sato-Tate Groups
  10. Explicit Maps to Lower Dimensional Jacobians
  11. Sato-Tate groups for curves of the form $y^2=x^{2^d+1}-x$
  12. Preliminaries
  13. Sato-Tate group results
  14. Sato-Tate groups for curves of the form $y^2=x^{2^m}-1$
  15. Preliminaries
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $m\geq3$ and $d\geq 1$ be integers. Then the Jacobians of the curves $C_{2^m}: y^2=x^{2^m} -1$ and $C'_{2^{d}+1}: y^2=x^{2^{d}+1}-x$ and their twists are nondegenerate.

Theorems & Definitions (32)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Conjecture 2.1: Algebraic Sato-Tate Conjecture
  • Proposition 2.2
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • ...and 22 more