Abstract
The groundfield k is algebraically closed and of characteristic p ≠ O. The p-rank of an abelian variety A/k is σA if there are σA copies of Z/pZ in the group of points of order p in A(k). The p-rank σX of a curve X/k is the p-rank of its Jacobian. In general the genus of X is ≥ σX. X is ordinary if equality holds.
Proposition 3.2 proves that the Artin-Schreier curve Xp with equation (xp−x)(yp−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 (σY−1+n)=(σX−1+n)×p. The particular case n=0, the unramiried case, is due to Šafarevič [7].
Similar content being viewed by others
References
Altman, A., Kleiman, S.L., Introduction to Grothendieck duality theory, Springer Lecture notes no 146(1970).
Frey G., Geyer, W.D., Über die fundamentalgruppe von korpern mit divisoren theorie, J. Reine Angew Math vol. 254, pp 110–122 (1972).
Hasse Helmut, und Witt Ernst, Zyklische unverzweigte erweiterungskorper von primzahlgrad p uber einen algebraischen funktionenkorper der characteristic p, Monatsh. Math. vol 43 pp 477–492 (1936).
Hurwitz, A., Analytische Gebilde mit eindeutigen trans formationen in sich, Math. Werke vol. 1, 391–430.
Miller, Leonhard, Curves with invertible Hasse-Witt matrix, Math. Ann. vol. 197, pp. 123–127 (1972).
Mumford, D., Abelian Varieties, Oxford University Press (1970).
Šafarevič, I.R., On p-Extensions, Amer. Math. Soc. Trnsla Series II, vol. 4, pp. 59–71 (1954).
Serre, J.P., Cohomologie Galoisienne, Springer Lecture notes no. 5, (1965).
Serre, J.P., Groupes Algébrique et Corps de Classes, Hermann, Paris, (1959).
Serre, J.P., Sur les corps Locaux à corps résiduel algébriquement clos., Bull. Soc. Math. France vol. 89, pp. 105–154, (1961).
Serre J.P., Sur la Topologie des Variétés Algébrique en caracteristic p., Mexico Symposium on Algebraic Topology, University of Mexico, pp. 24–53(1956).
Serre, J.P., L'Operation de Cartier (Private Notes at College de France, (1956)).
Author information
Authors and Affiliations
Additional information
The preparation of this paper was supported by the Memorial University of Newfoundland and NRC Grant A-8777.
Rights and permissions
About this article
Cite this article
Subrao, D. The p-rank of Artin-Schreier curves. Manuscripta Math 16, 169–193 (1975). https://doi.org/10.1007/BF01181639
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01181639