Explicit Families of Hyperelliptic Curves with CM Jacobians

TL;DR

This paper constructs explicit families of hyperelliptic curves over the rationals with Jacobians that have complex multiplication, using Chebyshev polynomials and Galois coverings to identify CM-fields and ensure simplicity.

Contribution

It introduces new explicit constructions of hyperelliptic curves with CM Jacobians over a0a0, detailing their defining equations and associated CM-fields.

Findings

Jacobians are proven to be simple

Explicit CM-fields are determined

Families are constructed using Chebyshev polynomials

Abstract

We construct explicit families of hyperelliptic curves over $\QQ$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by \[ v^2 = (u+2)\,\varphi_d(u), \quad d = 2^e \text{ or } d=p \geq 3 \text{ prime}, \] where $\varphi_d(x)$ is the Chebyshev polynomial of degree $d$. We prove that the Jacobians are simple and determine the associated CM-fields explicitly. Our approach exploits the interplay between Chebyshev polynomials and Galois coverings, providing concrete examples of abelian varieties with CM and explicit criteria for their construction.