Hasse-Witt matrices of Fermat curves

Abstract

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 x^m+y^m=a(≠0, a∈k) and x^m+y^m=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′).