Hyperelliptic parametrizations of $\mathbb{Q}$-curves

TL;DR

This paper introduces a method to compute urves over ields parametrized by rational points on hyperelliptic modular curves, expanding computational techniques in algebraic geometry and number theory.

Contribution

It provides a novel procedure for computing urves parametrized by rational points on hyperelliptic modular curves $X_0^*(N)$ for square-free integers $N$, specifically when these curves are hyperelliptic.

Findings

Developed an explicit computational procedure for urves

Applied the method to specific hyperelliptic modular curves

Enhanced understanding of urves parametrization on modular curves

Abstract

For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.