The p-rank of Artin-Schreier curves

Abstract

The groundfield k is algebraically closed and of characteristic p ≠ O. The p-rank of an abelian variety A/k is σ_A if there are σ_A copies of Z/pZ in the group of points of order p in A(k). The p-rank σ_X of a curve X/k is the p-rank of its Jacobian. In general the genus of X is ≥ σ_X. X is ordinary if equality holds.

Proposition 3.2 proves that the Artin-Schreier curve X_p with equation (x^p−x)(y^p−y)=1 is ordinary. As its genus is (p−1)(p−1) and it has at least 2p. p. (p−1) automorphisms, it is an ordinary counter example of Hurwitz's theorem if p>37. Theorem 3.5 is the inductive step in extending this to smaller characteristics. Both are corollaries of Theorem 4.1 which is our principal result: if Y→X is a cyclic covering of degree p ramified at n distinct points, then (σ_Y−1+n)=(σ_X−1+n)×p. The particular case n=0, the unramiried case, is due to Šafarevič [7].