Weierstrass points on cyclic covers of the projective line
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- by Christopher Towse PDF
- Trans. Amer. Math. Soc. 348 (1996), 3355-3378 Request permission
Abstract:
We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form $y^{n}=f(x)$, where $f$ is a polynomial of degree $d$. Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, $BW$. We obtain a lower bound for $BW$, which we show is exact if $n$ and $d$ are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula): \begin{equation*} \lim _{d\to \infty }\frac {BW}{g^{3}-g}=\frac {n+1}{3(n-1)^{2}}, \end{equation*} where $g$ is the genus of the curve. In the case that $n=3$ (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes $p$, the branch points and the non-branch Weierstrass points remain distinct modulo $p$.References
- Robert D. M. Accola, On generalized Weierstrass points on Riemann surfaces, Modular functions in analysis and number theory, Lecture Notes Math. Statist., vol. 5, Univ. Pittsburgh, Pittsburgh, PA, 1983, pp. 1–19. MR 732958
- Accola, R., Topics in the Theory of Riemann Surfaces, Lecture Notes in Mathematics, vol. 1595, Springer-Verlag, Berlin, Heidelberg, 1994.
- Jean-François Burnol, Weierstrass points on arithmetic surfaces, Invent. Math. 107 (1992), no. 2, 421–432. MR 1144430, DOI 10.1007/BF01231896
- M. Coppens, The Weierstrass gap sequences of the total ramification points of trigonal coverings of $\textbf {P}^1$, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 3, 245–276. MR 814880, DOI 10.1016/1385-7258(85)90039-3
- Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745, DOI 10.1007/978-1-4684-9930-8
- Hurwitz, A., Über algebraische Gebilde mit eindeutigen Transformationen in sich, Mathematische Annalen 41 (1893), 403–442.
- Takao Kato, On Weierstrass points whose first nongaps are three, J. Reine Angew. Math. 316 (1980), 99–109. MR 581326, DOI 10.1515/crll.1980.316.99
- Ja Kyung Koo, On holomorphic differentials of some algebraic function field of one variable over $\textbf {C}$, Bull. Austral. Math. Soc. 43 (1991), no. 3, 399–405. MR 1107394, DOI 10.1017/S0004972700029245
- Joseph Lewittes, Automorphisms of compact Riemann surfaces, Amer. J. Math. 85 (1963), 734–752. MR 160893, DOI 10.2307/2373117
- David Mumford, Curves and their Jacobians, University of Michigan Press, Ann Arbor, Mich., 1975. MR 0419430
- Hans Rademacher and Emil Grosswald, Dedekind sums, The Carus Mathematical Monographs, No. 16, Mathematical Association of America, Washington, D.C., 1972. MR 0357299, DOI 10.5948/UPO9781614440161
- Joseph H. Silverman, Some arithmetic properties of Weierstrass points: hyperelliptic curves, Bol. Soc. Brasil. Mat. (N.S.) 21 (1990), no. 1, 11–50. MR 1139554, DOI 10.1007/BF01236278
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Additional Information
- Christopher Towse
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003
- Email: towse@math.lsa.umich.edu
- Received by editor(s): September 27, 1994
- Received by editor(s) in revised form: October 16, 1995
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 3355-3378
- MSC (1991): Primary 14H55, 11G30
- DOI: https://doi.org/10.1090/S0002-9947-96-01649-2
- MathSciNet review: 1357406