Hyperelliptic Curves in Characteristic 2

Abstract

In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and 2-rank zero, given by an affine equation y^2-y=c_{2g+1} x^{2g+1} +...+ c_1 x. We prove that the first slope of the Newton polygon of X is bigger than or equal to 1/h. We also prove that the equality holds if (I) g<2^h-2, c_{2^h-1} is nonzero; or (II) g=2^h-2, c_{2^h-1} or c_{3(2^{h-1})-1} is nonzero. We prove that genus-4 hyperelliptic curve over F_2bar are precisely those with equations y^2 - y = x^9 + a x^5 + b x^3.