Tetsuo Kodama

Maximal hyperelliptic curves of genus three

This note contains general remarks concerning finite fields over which a so-called maximal, hyperelliptic curve of genus 3 exists. Moreover, the geometry of some specific hyperelliptic curves of genus 3 arising as quotients of Fermat curves, is studied. In particular, this results in a description of the finite fields over which a curve as studied here, is maximal.

Hasse-Witt matrices of Fermat curves

Let m be an integer with m≧3. Let K and K′ be perfect fields of characteristic p and p′ such that (p,m)=1 and (p′,m)=1, respectively. Moreover let A and A′ be algebraic function fields over K and K′ defined by xm+ym=a(≠0, a∈k) and xm+ym=a′(a′≠0 a′∈k′), respectively. Put g=(m−1)(m−2)/2. Denote by M(K,p,a) and M(K′,p′,a′) the Hasse-Witt matrices of A and A′ with respect to the canonical bases of holomorphic differentials. Then we show that if p+p′≡0(mod.m) then rank M(K,p,a)+rank M(K′,p′,a′)=g and if pp′≡1 (mod.m) then rank M(K,p,a)=rank M(K′,p′,a′).

A family of hyperelliptic function fields with Hasse-Witt-invariant zero

Let F = GF(q) be a finite field of characteristic p > 2. Let g be a positive integer. Denote by P(x) a polynomial over F of the form P(x) = x2g+1 + a, where (2g + 1, p) = 1 and a ∈ F×, or of the form P(x) = x(x2g + a), where (2g, p) = 1 and a ∈ F×. Let K = F(x, y) be a hyperelliptic function field defined by γ2 = P(x) over F. Put L(u) = (1 − γ1q12u) (1 − γ2q12u) ⋯ (1 − γ2gq12u) the L-function of K. Then, it is shown that, in the case P(x) = x2g+1 + a, γ1, γ2, …, γ2g are roots of unity if and only if there exists n ∈ N such that pn ≡ −1 (mod 4g + 2), and in the case P(x) = x(x2g + a), γ1, γ2, …, γ2g are roots of unity if and only if there exists n ∈ N such that pn ≡ −1 or 1 + 2g (mod 4g).