Arch. Math., Vol. 66, 150-156 (1996) 0003-889X/96/6602-0~50 $ 2.90/0 9 1996 Birkhguser Verlag, Basel Galois module structure of holomorphic differentials in characteristic p By MARTHA RZEDOWSKI-CALDERON, GABRIELVILLA-SALVADOR and MANOHAR L. MADAN 1. Introduction. Let K/k be a field of algebraic functions of one variable, k algebraically closed. If L/k is a finite Galois extension of K/k, the set s L of holomorphic differentials of L is a canonical k [G]-module, G = Gal (L/K). What is the explicit structure of this module in terms of indemcomposable k [G]-modules? In the classical case, this question was completely answered by Chevalley and Weil [2]. Hecke had suggested that one should study this problem "mit den Methoden der Algebra, unter Hervorhebung des Begriffes der Galoisschen Gruppe". In characteristic p, this is an interesting open problem. Kani [5] generalized the results of Chevalley and Weit to tamely ramified extensions. Special cases had been settled by Nakajima [6], Valentini [7] and Valentini and Madan [8]. For wildly ramified extensions the problem has been solved for cyclic p-extensions [8]. For non-cyclic extensions, the answer is not known even for the simplest case of the composite of two cyclic extensions of degree p! In this note we determine completely the structure of f2L as k [G]-module for a wide class of elementary abelian p-extensions of k (x). This class has been studied earlier by Garcia [3] and Garcia and Stichtenoth [4]. The statement of our result is quite similar to that of the main result of [8]. The key idea is to modify the Boseck-Garcia basis to make it convenient for analyzing the Galois action. 2. Holomorphie differentials. Let K = k (x), k an algebraically closed field of character- istic p > 0. Let L = K(y) where yq- y= (x - al) ..... (x - at) mr with q = p", n > l, g (x) ~ k [x], deg 9 (x) < ml + "'" + m, = m and mi relatively prime to p for i = 1,.-., r. Then L/K is a Galois extension with Galois group G = Gal(L/K) (IFq, +), lFq the finite field with q elements [3]. The isomorphism follows from the equality yq- y= • (Y+fl), fl~Fq so that for any fl ~Fq we have the element era ~ G given by ap(y) = y + ft. The primes P1 ..... Pr in K, P~ corresponding to (x - ai), are precisely the ramified primes and they are