Period Computations for Covers of Elliptic Curves

TL;DR

This paper develops a method to compute algebraic equations for curves covering elliptic curves, using period relations to approximate coefficients and verify their algebraic nature.

Contribution

It introduces a novel approach combining period relations and high-precision approximations to explicitly construct such covers and confirm their algebraic coefficients.

Findings

Derived algebraic equations for covers of elliptic curves.

Proved the conjecture about the algebraic nature of the coefficients.

Provided a computational framework for similar problems.

Abstract

In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these relations to approximate the coefficients to high precision. We then conjecture which algebraic numbers the coefficients are, and then we prove this conjecture to be correct.