On Effective Sato-Tate Distributions for Surfaces Arising from Products of Elliptic Curves

TL;DR

This paper establishes effective Sato-Tate distribution results for certain surfaces derived from products of elliptic curves, including K3 and double quadric surfaces, with explicit error bounds.

Contribution

It provides the first effective error bounds for Sato-Tate distributions in these surface families, including cases with CM elliptic curves.

Findings

Unconditional effective error bounds for Sato-Tate distributions.

Effective joint distribution for two twist-inequivalent elliptic curves.

Extension of previous work to include CM elliptic curves.

Abstract

We prove, with an unconditional effective error bound, the Sato-Tate distributions for two families of surfaces arising from products of elliptic curves, namely a one-parameter family of K3 surfaces and double quadric surfaces. To prove these effective Sato-Tate distributions, we prove an effective form of the joint Sato-Tate distribution for two twist-inequivalent elliptic curves, along with an effective form of the Sato-Tate distribution for an elliptic curve for primes in arithmetic progressions. The former completes the previous work of Thorner by including the cases in which one of the elliptic curves has CM.