Artin-Schreier curves of prescribed p-rank without extra automorphisms

Source abstract

Let p be an odd prime. We study the Artin-Schreier analogue of Poonen's problem of constructing curves without extra automorphisms. An Artin-Schreier curve carries an unavoidable deck group Cp, so the smallest possible full geometric automorphism group inside the Artin-Schreier locus is Cp. We determine the exact numerical range in which this minimum is attained, for every p-rank. More precisely, for integers r >= 0 and N >= 1, there exists an Artin-Schreier curve X/Fp with g(X) = (p - 1)N/2, gamma(X) = r(p - 1), Aut_Fp(X) is isomorphic to Cp if and only if either r = 0, p does not divide N + 1, N >= 3 and (p,N) not equal to (3, 3), or r >= 1, N >= 2r and (r,N) not equal to (1, 2). All models are defined over the prime field by explicit equations together with elementary choices of auxiliary polynomials, and every excluded admissible case is shown to carry an extra geometric automorphism. The main structural input is the theorem of Valentini and Madan on the uniqueness of the rational subfield of index p outside three exceptional families. We exclude these families by intrinsic arguments: the genus and p-rank in most cases, Nakajima's bound on p-subgroups of automorphism groups of ordinary curves for the ordinary family, and the Lehr-Matignon normal form for isomorphisms of one-point covers for the monomial family.