Real multiplication curves by subfields of cyclotomic fields

TL;DR

This paper constructs explicit families of algebraic curves with real multiplication by subfields of cyclotomic fields, building on Ellenberg's theoretical existence results.

Contribution

It provides explicit examples of real multiplication curves for all cases previously shown to exist by Ellenberg.

Findings

Explicit families of real multiplication curves are constructed.

The work confirms the existence of these curves in concrete form.

Provides tools for further research in algebraic geometry and number theory.

Abstract

In \emph{Endomorphism Algebras of Jacobians}, Ellenberg gives group theory tools to construct jacobians of curves with real multiplication. He shows the existence of curves and family of curves with real multiplication by subfields of cyclotomic fields. Among them, some are already known such as the family of Mestre, or the family of Tautz, Top and Verberkmoes. In this article, we give explicit families of real multiplication curves, for each case Ellenberg showed their existence.