Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx

TL;DR

This paper investigates the distribution of Frobenius traces for specific genus 3 hyperelliptic curves with large automorphism groups, providing algorithms, heuristic descriptions, and proofs of their Sato-Tate groups.

Contribution

It introduces efficient algorithms for computing Frobenius traces and confirms heuristic Sato-Tate group descriptions through explicit calculations and Galois endomorphism types.

Findings

Heuristic descriptions of Sato-Tate groups are validated.

Algorithms enable efficient Frobenius trace computations.

Explicit Sato-Tate groups are determined for the families.

Abstract

We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We then prove that these heuristic descriptions are correct by explicitly computing the Sato-Tate groups via the correspondence between Sato-Tate groups and Galois endomorphism types.