Explicit Families of Hyperelliptic Curves with CM Jacobians
Saeed Tafazolian, Jaap Top
TL;DR
The paper constructs explicit hyperelliptic curves over ${\mathbb{Q}}$ whose Jacobians have complex multiplication by explicit CM-fields. It uses Chebyshev polynomials to define curves ${\mathcal{C}}_d: v^2=(u+2)\varphi_d(u)$ and analyzes them via Galois coverings to produce explicit endomorphisms, validating CM by $K_d=\mathbb{Q}(\zeta_d-\zeta_d^{-1})$ (for $d=2^e$) or $K_d=\mathbb{Q}(\zeta_p)$ (for odd prime $d=p$). The authors prove the Jacobians are simple by establishing primitive CM-types and compute the corresponding CM-fields concretely, leveraging tangent-space reasoning and automorphism actions on regular differentials. The results provide concrete, number-theoretically rich families of CM abelian varieties defined over ${\mathbb{Q}}$, with explicit CM-types and endomorphism structures that illuminate the interplay between Chebyshev polynomials, Galois coverings, and CM theory.
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.