First vertices for hyperelliptic curves in characteristic two

TL;DR

This paper investigates the Newton polygons of zeta functions for hyperelliptic curves of 2-rank 0 in characteristic two, identifying their initial vertices in generic and some special cases.

Contribution

It determines the first vertices of Newton polygons for these hyperelliptic curves, advancing understanding of their arithmetic properties in characteristic two.

Findings

Identified the first generic vertex of the Newton polygon.

Determined the first vertex in certain non-generic cases.

Enhanced understanding of zeta functions for hyperelliptic curves in characteristic two.

Abstract

We study the Newton polygons of numerators of the zeta functions of $2$-rank $0$ hyperelliptic curves in characteristic $2$. We determine their first generic vertex, and their first vertex in some other non generic cases.