The existence of supersingular curves of genus 4 in arbitrary characteristic
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RESEARCH
The existence of supersingular curves of genus 4 in arbitrary characteristic Momonari Kudo1 , Shushi Harashita2* * Correspondence:
[email protected] Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan Full list of author information is available at the end of the article 2
and Hayato Senda3
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 P1 of two supersingular elliptic curves is supersingular.
1 Introduction 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. As to supersingular curves, the following is a basic problem (cf. [19, Question 2.2]). For given g, does there exist a supersingular curve of genus g in any characteristic p? For g ≤ 3, this problem was solved affirmatively. The case of g = 1, i.e., elliptic curves is due to Deuring [4]. Ibukiyama, 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]. Also, 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 (cf. Serre [20, Théorème 3]) and if g = 3, p ≥ 3 and e is odd (cf. Ibukiyama [10, Theorem 1]), where we recall the general fact that any maximal curve over Fp2 is superspecial (and therefore supersingular). Even for (g, p) = (2, 3), there exists a supersingular curve: for example y2 = x5 + 1 is supersingular (but is not superspecial), since its Cartier-Manin matrix is nilpotent, see (9) and (10) in Sect. 2 for the Cartier–Manin matrix and a criterion for the supersingularity. 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. This paper focuses on the first open case, i.e., the case of g = 4 (cf. [19, Question 3.4]). Let us recall some recent works, restricting ourselves to the case of g = 4. According to [16, Remark 7.2] and [15, Theorem 7.1] by Li, Mantovan, Pries and Tang, there exists a supersingular curve of genus 4 if p ≡ 5 mod 6 or if p ≡ 2, 3, 4 mod 5. Among them, we review the existence for (g, p) = (4, 3) for the reader’s convenience. Indeed y10 = x(1 − x)
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