Explicit supersingular cyclic curves

TL;DR

This paper explicitly constructs supersingular cyclic curves over algebraic closures of finite fields by leveraging reductions of complex multiplication curves, building on prior existence proofs.

Contribution

It makes explicit certain supersingular curves previously known only to exist through non-constructive proofs, using reductions of CM curves.

Findings

Explicit equations for supersingular cyclic curves obtained

Connections established between CM curves and supersingular curves

Enhances understanding of supersingular curve classifications

Abstract

Li, Mantovan, Pries, and Tang proved the existence of supersingular curves over Fpbar in each of the special families of curves in Moonen's classification. Their proof does not provide defining equations of these curves. We make some of their results explicit using the reductions modulo p of previously computed curves with complex multiplication.