Hyperelliptic Curves in Characteristic 2

In this paper we prove that there are no hyperelliptic supersingular curves of genus 2n − 1 in characteristic 2 for any integer n ≥ 2. Let F be an algebraically closed field of characteristic 2, and let g be a positive integer. Write h = blog2(g + 1) + 1c, where b c denotes the greatest integer less than or equal to a given real number. Let X be a hyperelliptic curve over F of genus g ≥ 3 of 2-rank zero, given by an affine equation y2 − y = ∑2g+1 i=1 cix i. We prove that the first slope of the Newton polygon of X is ≥ 1/h. We also prove that the equality holds if (I) g < 2h − 2, c2h−1 6= 0; or (II) g = 2h − 2, c2h−1 6= 0 or c3·2h−1−1 6= 0. Let HSg/F be the intersection of the supersingular locus with the open hyperelliptic Torelli locus in the moduli space of principally polarized abelian varieties over F of dimensions g. We show that dimHSg/F ≤ g − 2 for every g ≥ 3. We prove that dimHS4/F = 2 by showing that every genus-4 hyperelliptic supersingular curve over F has an equation y2−y = x+c5x+c3x for some c5, c3 ∈ F .