Supersingular genus-two curves over fields of characteristic three

TL;DR

This paper classifies certain degree-3 maps from supersingular genus-2 curves over algebraically closed fields of characteristic 3 to elliptic curves with j-invariant 0, and determines Weil polynomials over finite fields.

Contribution

It establishes the exact number of degree-3 maps for most supersingular genus-2 curves and analyzes the moduli space, advancing understanding of their rationality and Weil polynomials.

Findings

Exactly 20 degree-3 maps from C to E for most curves

Classification of the moduli space of triples (C, E, phi)

Determination of Weil polynomials over finite fields of characteristic 3

Abstract

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the elliptic curve E with j-invariant 0. We study the coarse moduli space of triples (C,E,phi), paying particular attention to questions of rationality. The results we obtain allow us to determine, for every finite field k of characteristic 3, the polynomials that occur as Weil polynomials of supersingular genus-2 curves over k.