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Degeneracy and Sato-Tate groups of $y^2=x^{p^2}-1$

Justin Chen, Heidi Goodson, Rezwan Hoque, Sabeeha Malikah

TL;DR

This work treats the degenerate Jacobians in the CM family C_{p^2} and furnishes an explicit description of their Sato-Tate groups. By applying Shioda's Hodge-class construction, the authors classify indecomposable Hodge classes and derive the identity component ST^0 ≅ U(1)^{g'} and a cyclic component group of order φ(p^2), with an explicit Galois-driven generator γ. The results give a complete account of how degeneracy alters the Sato-Tate structure for an infinite family of CM abelian varieties, and they illustrate the approach with detailed moment statistics for the case p^2 = 25. The methodology combines CM theory, Hodge theory, and computational techniques to yield explicit descriptions of both the Sato-Tate group and its statistical behavior, with potential applicability to broader families exhibiting degeneracy.

Abstract

We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves $C_{p^2}: y^2=x^{p^2}-1$, where $p$ is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper.

Degeneracy and Sato-Tate groups of $y^2=x^{p^2}-1$

TL;DR

This work treats the degenerate Jacobians in the CM family C_{p^2} and furnishes an explicit description of their Sato-Tate groups. By applying Shioda's Hodge-class construction, the authors classify indecomposable Hodge classes and derive the identity component ST^0 ≅ U(1)^{g'} and a cyclic component group of order φ(p^2), with an explicit Galois-driven generator γ. The results give a complete account of how degeneracy alters the Sato-Tate structure for an infinite family of CM abelian varieties, and they illustrate the approach with detailed moment statistics for the case p^2 = 25. The methodology combines CM theory, Hodge theory, and computational techniques to yield explicit descriptions of both the Sato-Tate group and its statistical behavior, with potential applicability to broader families exhibiting degeneracy.

Abstract

We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves , where is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper.

Paper Structure

This paper contains 14 sections, 23 theorems, 54 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Background on degeneracy
  4. Computation of the Sato-Tate group
  5. Preliminaries on the curve $C_{p^2}$ and its Jacobian
  6. Endomorphisms of the Jacobian
  7. Degeneracy
  8. General degeneracy results
  9. Shioda's construction
  10. Complete description of the indecomposable Hodge classes when $m=p^2$
  11. The Sato-Tate Group of $\mathop{\mathrm{Jac}}\nolimits(C_{p^2})$
  12. Moment Statistics
  13. Computing theoretical moments
  14. Example: $p^2=25$

Key Result

Lemma 2.1

We have the following isogeny over $\mathbb{Q}$: where $C_p/\mathbb{Q}$ denotes the curve $y^2=x^p-1$ and $X_{p^2}$ is a simple abelian variety of dimension $\phi(p^2)/2$ with complex multiplication by $\mathbb{Q}(\zeta_{p^2})$.

Figures (1)

  • Figure 1: Histogram of the $a_1$-coefficients for $C_{25}: y^2=x^{25}-1$.

Theorems & Definitions (52)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • ...and 42 more