Jacobians with complex multiplication

TL;DR

This paper constructs specific algebraic curves with Jacobians that admit complex multiplication, using Galois coverings with particular metacyclic groups, and analyzes their properties and CM-types.

Contribution

It introduces new families of curves with Jacobians having complex multiplication, derived from Galois coverings with metacyclic groups, and characterizes their CM-types and simplicity.

Findings

Jacobians admit complex multiplication from constructed curves

Jacobians are shown to be simple abelian varieties

Explicit CM-types are determined for the Jacobians

Abstract

We construct and study two series of curves whose Jacobians admit complex multiplication. The curves arise as quotients of Galois coverings of the projective line with Galois group metacyclic groups $G_{q,3}$ of order $3q$ with $q \equiv 1 \mod 3$ an odd prime, and $G_m$ of order $2^{m+1}$. The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the CM-types and show that the Jacobians are simple abelian varieties.