Abstract
We prove that there exists a supersingular nonsingular curve of genus 4 in arbitrary characteristic p>0. For p>3 we shall prove that the desingularization of a certain fiber product over mathbf{P }^1 of two supersingular elliptic curves is supersingular.
Highlights
Let K be an algebraically closed field of positive characteristic
Does there exist a supersingular curve of genus g in any characteristic p?
As a proof for g = 2 with p > 3 and for g = 3 with p > 2, we refer to the stronger fact that there exists a maximal curve of genus g over Fp2e if g = 2 and p2e = 4, 9
Summary
Let K be an algebraically closed field of positive characteristic. For a nonsingular algebraic curve C over K we call C supersingular (resp. superspecial) if its Jacobian J (C) is isogenous (resp. isomorphic) to a product of supersingular elliptic curves. Does there exist a supersingular curve of genus g in any characteristic p?. Katsura and Oort in [11, Proposition 3.1] proved the existence of superspecial curves of genus 2 for p > 3. For the existence of supersingular curves of genus 3 in any characteristic p > 0, see Oort [17, Theorem 5.12]. For the case of p = 2, we refer to the celebrated paper [22] by van der Geer and van der Vlugt, where they proved that there exists a supersingular curve of an arbitrary genus in characteristic 2.
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