Abstract
We consider hyperelliptic curves in characteristic p>2 for which the Hasse-Witt matrix is the zero matrix. For such curves, we establish an upper bound on the genus g, namely g≤(p-1)/2. For g=(p-1)/2, we establish the fact that up to isomorphism, there is precisely one such curve, namely the one given by the equation y2=xp-x. We determine all hyeprelliptic curves of the form y2=xn-x or y2=xn-1 for which the Hasse-Witt matrix is the zero matrix.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ekedahl, T.: On supersingular curves and abelian varieties. Math. Scand.60 151–178 (1987)
Husemöller, D.: Elliptic Cruves. Berlin-Heidelberg-New York: Springer-Verlag 1987
Kodama, T. and Washio, T.: A, Family of Hyperelliptic Function Fields with Hasse-Witt Invariant Zero. J. Number Theory36 187–200
Nygaard, N.: Slpes of powers of froebenius on crystalline cohomology. Ann. Sci. École Norm. Sup.14 369–401 (1981)
Yui, N.: On the Jacobian Varieties of Hyperelliptic Curves over Fields of Characteristic p>2. J. Algebra52 378–410 (1978)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Valentini, R.C. Hyperelliptic curves with zero Hasse-Witt matrix. Manuscripta Math 86, 185–194 (1995). https://doi.org/10.1007/BF02567987
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02567987