Explicit Families of Hyperelliptic Curves with CM Jacobians

We construct explicit families of hyperelliptic curves over $\text{QQ}$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by $$v^2=(u+2){\phi }_d(u),d=2^e\text{ or }d=p\ge 3\text{ prime},$$ where ${\phi }_d(x)$ 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.