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Abelian surfaces over finite fields containing no curves of genus $3$ or less

Elena Berardini, Alejandro Giangreco Maidana, Stefano Marseglia

TL;DR

The paper investigates abelian surfaces over finite fields that contain no curves of genus up to $3$, culminating in several intertwined classifications. It provides a complete description of isogeny classes with no curves of genus $\le 2$ via Honda–Tate Weil polynomials, and proves a central equivalence: for simple abelian surfaces, containing a genus-$3$ curve is equivalent to possessing a polarisation of degree $4$, enabling algorithmic identification of such classes. Using Howe’s kernel framework, it derives necessary and sufficient conditions for the existence of degree-$4$ polarisations and connects these to the detailed arithmetic of the CM-field $K$ determined by the Weil polynomial. The authors also show that absolutely irreducible genus-$3$ curves on these surfaces have point counts far from the Serre–Weil bound and provide computational tools and examples (via Mar21 and LMFDB) to enumerate isomorphism classes with degree-$4$ polarisations. Overall, the work advances both the theoretical understanding and practical computation of genus-constraints on abelian surfaces over finite fields, with implications for coding theory and arithmetic geometry.

Abstract

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to $2$ initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus $3$ is equivalent to admitting a polarisation of degree $4$. Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus $2$ curves have a polarisation of degree $4$. Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus $\leq 2$, containing no abelian surface with a polarisation of degree $4$. Finally, we describe the absolutely irreducible genus $3$ curves lying on abelian surfaces containing no curves of genus less than or equal to $2$, and show that their number of rational points is far from the Serre--Weil bound.

Abelian surfaces over finite fields containing no curves of genus $3$ or less

TL;DR

The paper investigates abelian surfaces over finite fields that contain no curves of genus up to , culminating in several intertwined classifications. It provides a complete description of isogeny classes with no curves of genus via Honda–Tate Weil polynomials, and proves a central equivalence: for simple abelian surfaces, containing a genus- curve is equivalent to possessing a polarisation of degree , enabling algorithmic identification of such classes. Using Howe’s kernel framework, it derives necessary and sufficient conditions for the existence of degree- polarisations and connects these to the detailed arithmetic of the CM-field determined by the Weil polynomial. The authors also show that absolutely irreducible genus- curves on these surfaces have point counts far from the Serre–Weil bound and provide computational tools and examples (via Mar21 and LMFDB) to enumerate isomorphism classes with degree- polarisations. Overall, the work advances both the theoretical understanding and practical computation of genus-constraints on abelian surfaces over finite fields, with implications for coding theory and arithmetic geometry.

Abstract

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to . Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus is equivalent to admitting a polarisation of degree . Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus curves have a polarisation of degree . Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus , containing no abelian surface with a polarisation of degree . Finally, we describe the absolutely irreducible genus curves lying on abelian surfaces containing no curves of genus less than or equal to , and show that their number of rational points is far from the Serre--Weil bound.
Paper Structure (7 sections, 39 theorems, 57 equations, 1 figure)

This paper contains 7 sections, 39 theorems, 57 equations, 1 figure.

Table of Contents

  1. Abelian surfaces containing no curves of genus <= 2
  2. Abelian surfaces containing genus 3 curves
  3. Factorisation of 2
  4. Kernels of polarisations of abelian varieties
  5. Polarisations of degree 4 on abelian surfaces
  6. Computing isomorphism classes admitting a polarisation of degree 4
  7. Genus 3 curves lying on abelian surfaces

Key Result

Corollary 1

Let $f(t)$ be the Weil polynomial of a simple isogeny class $\mathcal{C}$ of abelian surfaces over $\mathbb{F}_q$.

Figures (1)

  • Figure 1: Possible factorisations of the prime $2$ in $K/K^+/\mathbb{Q}$.

Theorems & Definitions (88)

  • Corollary
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Proposition 1.3
  • proof
  • Theorem 1.4
  • proof
  • Remark 1.5
  • ...and 78 more