Hyperelliptic curves in characteristic 2

2000/12/31 by Jasper Scholten, Hui June Zhu · 1 citation
Mathematics · #math.AG #math.NT

paper · pdf

published as Inter. Math. Research Notices, 17 (2002), 905-917. · 10 pages

arxiv created 2001/05/03 · arxiv updated 2009/11/30

Abstract

In this paper we prove that there are no hyperelliptic supersingular curves over F₂bar of genus 2ⁿ-1 for any integer n>1. Let g be a natural number, and h=floor(log₂(g+1)+1). Let X be a hyperelliptic curve over F₂bar of genus g>2 and 2-rank zero, given by an affine equation y²-y=c_2g+1 x^2g+1 +...+ c₁ 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 F₂bar are precisely those with equations y² - y = x⁹ + a x⁵ + b x³.

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• An unusual family of supersingular curves of genus five

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