On the Existence of Supersingular Curves Of Given Genus

In this note we shall show that there exist supersingular curves for every positive genus in characteristic 2. Recall that an irreducible smooth algebraic curve C over an algebraically closed field F of characteristic p > 0 is called supersingular if its jacobian is isogenous to a product of supersingular elliptic curves. An elliptic curve is called supersingular if it does not have points of order p over F. It is not clear a priori that there exist such curves for every genus. Indeed, note that in the moduli space Ag ⊗ Fp of principally polarized abelian varieties the locus of supersingular abelian varieties has dimension [g/4] (cf. [O, L-O]), while the locus of jacobians has dimension 3g − 3 for g > 1. Therefore, as far as dimensions are concerned there is no reason why these loci should intersect for g ≥ 9. In this paper we construct for every integer g > 0 a supersingular curve of genus g over the field F2. In particular this shows that for every g > 0 there exists an irreducible curve of genus g whose jacobian is isogenous to a product of elliptic curves. We refer to [E-S] for related questions in characteristic 0. We do our construction by taking a suitable fibre product of Artin-Schreier curves. This construction is inspired by coding theory, where the introduction of generalized Hamming weights led us to consider such products, cf. [G-V 2]. More generally, we are able to construct in characteristic p a supersingular curve over Fp of any genus g whose p-adic expansion consists of the digits 0 and (p− 1)/2 only. We can also count on how many moduli the construction depends.