On Weil Polynomials of Hyperelliptic Curves over Finite Fields of Characteristic 2

TL;DR

This paper develops new conditions to identify when hyperelliptic Jacobians cannot exist in certain isogeny classes over finite fields of characteristic 2, and provides an efficient enumeration algorithm for hyperelliptic curves.

Contribution

It extends previous methods by analyzing Weil polynomial coefficients modulo 2, and introduces a practical enumeration algorithm for hyperelliptic curves over characteristic 2 fields.

Findings

Weil polynomial coefficients cannot have certain residues modulo 2 for Jacobians.

Asymptotic equidistribution of Weil coefficient parities in high dimensions.

Obstructions exclude a significant fraction of isogeny classes from containing hyperelliptic Jacobians.

Abstract

We present new conditions which obstruct the existence of hyperelliptic Jacobians in isogeny classes of abelian varieties over finite fields of characteristic 2. We show that Weil polynomials of Jacobians cannot have coefficients in certain residue classes modulo 2, extending the approach of Costa et al. in arXiv:2002.02067. We prove that for 3- and 4-dimensional abelian varieties over $\mathbb{F}_{2^n}$, as $n \rightarrow\infty$, the parities of the Weil coefficients asymptotically equidistribute. Further, we show that these obstructions disqualify $\frac12$ of all 3-dimensional isogeny classes and $\frac 58$ of all 4-dimensional isogeny classes from containing a hyperelliptic Jacobian. Additionally, we present a practical enumeration algorithm which generates all isomorphism classes of hyperelliptic curves of arbitrary genus over almost any finite field of characteristic 2 based on existing algorithms over $\mathbb{F}_2$. Our analysis shows the runtime to be $\tilde{{O}}(2^{n(2g-1)})$ expected, and $\tilde{{O}}(2^{n(2g+2)})$ worst case. This runtime improvement renders the algorithm practical for fields other than $\mathbb{F}_2$.