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Explicit supersingular cyclic curves

Marco Streng

TL;DR

This work makes explicit, over finite fields, the existence of supersingular curves in three Moonen families of cyclic covers by reducing CM curves modulo primes. It provides explicit models for M[6], M[8], and M[16], defined over $\mathbb{F}_p$ under clear congruence conditions, and strengthens previous results by eliminating the need for primes to be sufficiently large. The approach combines CM theory with concrete reductions, using CM fields like $\mathbb{Q}(\zeta_9)$ and related constructions, and adapts CM-based arguments to show supersingularity for almost all admissible primes, with certain cases reliant on conjectures. The results contribute concrete, computable families of supersingular curves within these special subvarieties, while highlighting the need for CM-curve databases and reconstruction algorithms to support broader verification and application.

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.

Explicit supersingular cyclic curves

TL;DR

This work makes explicit, over finite fields, the existence of supersingular curves in three Moonen families of cyclic covers by reducing CM curves modulo primes. It provides explicit models for M[6], M[8], and M[16], defined over under clear congruence conditions, and strengthens previous results by eliminating the need for primes to be sufficiently large. The approach combines CM theory with concrete reductions, using CM fields like and related constructions, and adapts CM-based arguments to show supersingularity for almost all admissible primes, with certain cases reliant on conjectures. The results contribute concrete, computable families of supersingular curves within these special subvarieties, while highlighting the need for CM-curve databases and reconstruction algorithms to support broader verification and application.

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.
Paper Structure (7 sections, 6 theorems, 12 equations)

This paper contains 7 sections, 6 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Supersingular Picard curves in M[6]
  3. Supersingular hyperelliptic curves in M[8]
  4. Supersingular cyclic plane quintics in M[16]
  5. Final notes
  6. Other families
  7. Databases of curves, and verification

Key Result

Theorem 1.1

In each of the following families there exists a supersingular smooth curve of genus $g$ defined over $\overline{\mathbb{F}_p}$ for all sufficiently large primes $p$ that satisfy the given condition:

Theorems & Definitions (14)

  • Theorem 1.1: li-mantovan-pries-tang
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Conjecture 4.1
  • ...and 4 more