Some arithmetic properties of Weierstrass points: Hyperelliptic curves
- Joseph H. Silverman
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
The set of Weierstrass points for pluricanonical linear systems on an algebraic curve C have been extensively studied from a geometric viewpoint. If the curve is defined over a number fieldk, then thesen th order Weierstrass points are defined over an algebraic extensionk n ofk, and it is an interesting question to ask for the arithmetic properties of the points and the extension that they generate. In this paper we begin the study of the arithmetic properties of higher order Weierstrass points in the special case of hyperelliptic curves. We give an upper bound for the average height of these points, and we show that for sufficiently large primesp, the first order Weierstrass points and then th order Weierstrass points remain distinct modulop. This limits to some extent the ramification that can occur in the extensionk n /k. We also present two numerical examples which indicate that a complete description of the ramification is likely to be complicated.
- Accola, R.,On generalized Weierstrass points on Riemann surfaces, “Modular Functions in Analysis and Number Theory, ed. by T.A. Metzger, Lecture Notes in Math. and Stat.,” Univ. of Pittsburgh, Pittsburgh, PA, 1978.
- Gessel, I., Viennot, G.,Binomial determinants, paths, and hook length formulae Advances in Math.58 (1985), 300–320.
- Griffiths, P., Harris, J., “Principles of Algebraic Geometry,” John Wiley & Sons, New York, 1978.
- Hartshorne, R., “Algebraic Geometry,” Springer, New York, 1977.
- Lang, S., “Fundamentals of Diophantine Geometry,” Springer, New York, 1983.
- Laskov, D.,Weierstrass points on curves, Astérisque87–88 (1981), 221–248.
- Mumford, D., Fogarty, J., “Geometric Invariant Theory,” 2 nd edition, Springer, Berlin, 1982.
- Mumford, D., “Curves and Their Jacobians,” Univ. of Mich. Press, Ann Arbor.
- Neeman, A.,The distribution of Weierstrass points on a compact Riemann surface, Annals of Math.120, 317–328.
- Neeman, A.,Weierstrass points in characteristic p, Invent. Math.75, 359–376.
- Rohrlich, D.,Some remarks on Weierstrass points, Number Theory Related to Fermat's Last Theorem, N. Koblitz, ed., Boston, Birkhäuser.
- Stöhr, K.-O. Voloch, J.F.,Weierstrass points and curves over finite fields, Proc. London Math. Soc.52, 1–19.
- van der Waerden, B.L., “Algebra,” Vol. 1, Fred. Ungar. Publ. Co., New York.
- Some arithmetic properties of Weierstrass points: Hyperelliptic curves
Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society
Volume 21, Issue 1 , pp 11-50
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Author Affiliations
- 1. Mathematics Department, Brown University, 02912, Providence, RI, USA