Abstract
The purpose of this paper is to describe some recent results regarding the arithmetic properties of Weierstrass points on modular curves X 0 (p) for primes p. We begin with some generalities; most of these can be found, for example, in the book of Farkas and Kra [F-K]. Suppose that X is a compact Riemann surface of genus g ≥2. If γ is a positive integer, then let H r (X) denote the space of holomorphic r-differentials on X. Each H r (X) is a finite-dimensional vector space over ℂ; we denote its dimension by d r (X). A point Q ∈ X is called an r-Weierstrass point if there exists a non-zero differential w ∈ H r (X) such that
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References
[A-S. Ahlgren and K. Ono, Weierstrass points on X o (p) and supersingular j-invariants, Math. Ann. 325 (2003), 355–368.
S. Ahlgren and M. Papanikolas, Higher Weierstrass points on X o (p), Trans. Amer. Math. Soc. 355 (2003), 1521–1535.
A. O. L. Atkin, Weierstrass points at cusps of X O (N), Ann. of Math. 85 (1967), 42–45.
H. M. Farkas and I. Kra, “Riemann surfaces,” Springer-Verlag, New York, 1992.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, in “Computational perspectives on number theory (Chicago, IL,1995),” Amer. Math. Soc., Providence, RI, 97–126, 1998.
J. Lehner and M. Newman, Weierstrass points on Γ0 (N), Ann. of Math. 79 (1964), 360–368.
A. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462.
A. Ogg, On the Weierstrass points of Xo(N), Illinois J. Math. 22 (1978), 31–35.
D. Rohrlich, Weierstrass points and modular forms, Illinois J. Math. 29 (1985), 134–141.
H. P. F. Swinnerton-Dyer, On P-adic representations and congruences for mod- ular forms, in “Modular functions of one variable, III,” 1–55, Lecture Notes in Mathematics, 350. Springer-Verlag, Berlin.
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© 2004 Kluwer Academic Publishers
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Ahlgren, S. (2004). The Arithmetic of Weierstrass Points on Modular Curves X 0 (p). In: Hashimoto, Ki., Miyake, K., Nakamura, H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0249-0_1
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DOI: https://doi.org/10.1007/978-1-4613-0249-0_1
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