Polynesian Journal of Mathematics

Volume 4, Issue 3
Special volume AJAX in honor of René Schoof

Explicit families of hyperelliptic curves with CM Jacobians

Saeed Tafazolian and Jaap Top

Abstract:

We construct explicit families of hyperelliptic curves over $\mathbb{Q}$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by

$v^2=(u+2){\phi }_d\left(u\right),d=2^e\text{ or }d=p\ge 3\text{ prime},$

where ${\phi }_d\left(x\right)$ is the Chebyshev polynomial of degree $d$. We prove that the Jacobians are absolutely 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.

Keywords: Complex multiplication, hyperelliptic curves, Jacobians, abelian varieties, CM-types, Chebyshev polynomials.

Citation:

Saeed Tafazolian and Jaap Top. Explicit families of hyperelliptic curves with CM Jacobians. Polynesian Journal of Mathematics, Volume 4, Issue 3, Pages 1–11. (Sept. 2026) DOI: 10.69763/polyjmath.4.3

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Milestones:

Received 11 Nov 2025
Revised 31 Mar 2026
Accepted 31 Mar 2026
Published 28 Sep 2026
Communicated by Elisa Lorenzo García

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