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Unramified abelian covers with many points

Jean Gasnier

TL;DR

The paper addresses constructing curves over finite fields with record numbers of rational points by exploiting unramified abelian covers. It uses Class Field Theory in the unramified setting to realize covers as pullbacks of isogenies via the Jacobi map, enabling a correspondence between subgroups of the rational points of the Jacobian $\\mathcal{J}_X(K)$ and the covers. By leveraging a quadratic base-change trick and LMFDB data for L-polynomials, the authors compute explicit curves with record point counts over $\\mathbb{F}_4$, $\\mathbb{F}_9$, $\\mathbb{F}_{16}$, and $\\mathbb{F}_{25}$, improving previous records from the manypoints database. These constructions provide new high-point curves with potential impact on algebraic-geometry codes and point-count optimization over small fields.

Abstract

We produce curves with a record number of points over the finite fields with $4$, $9$, $16$ and $25$ elements, as unramified abelian covers of algebraic curves.

Unramified abelian covers with many points

TL;DR

The paper addresses constructing curves over finite fields with record numbers of rational points by exploiting unramified abelian covers. It uses Class Field Theory in the unramified setting to realize covers as pullbacks of isogenies via the Jacobi map, enabling a correspondence between subgroups of the rational points of the Jacobian and the covers. By leveraging a quadratic base-change trick and LMFDB data for L-polynomials, the authors compute explicit curves with record point counts over , , , and , improving previous records from the manypoints database. These constructions provide new high-point curves with potential impact on algebraic-geometry codes and point-count optimization over small fields.

Abstract

We produce curves with a record number of points over the finite fields with , , and elements, as unramified abelian covers of algebraic curves.
Paper Structure (3 sections, 2 theorems, 16 equations, 1 figure, 4 tables)

This paper contains 3 sections, 2 theorems, 16 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Some recalls on abelian covers
  3. New record curves

Key Result

Theorem 1

Let $K$ be a finite field and let $X$ be a smooth projective curve over $K$. Let $P$ be a rational point of $X$. There exists a smooth projective curve $Y_{max}$ over $K$ and an abelian covering of $X$ that is unramified, totally split above $P$, and maximal in the following sense: for any unramified abelian covering satisfying these properties, there exists an unramified abelian covering such

Figures (1)

  • Figure 1: Unramified abelian covers of $X$ totally split above $P$

Theorems & Definitions (2)

  • Theorem 1: Serre84
  • Proposition 2