The Arithmetic of Weierstrass Points on Modular Curves X0 (p)

  • Scott Ahlgren
Part of the Developments in Mathematics book series (DEVM, volume 11)


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 wH r (X) such that
$${\operatorname{ord} _{Q\omega }} \geqslant {d_r}\left( X \right)$$


Modular Form Cusp Form Compact Riemann Surface Weierstrass Point Modular Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A-O]
    [A-S. Ahlgren and K. Ono, Weierstrass points on X o (p) and supersingular j-invariants, Math. Ann. 325 (2003), 355–368.MathSciNetMATHCrossRefGoogle Scholar
  2. [A-P]
    S. Ahlgren and M. Papanikolas, Higher Weierstrass points on X o (p), Trans. Amer. Math. Soc. 355 (2003), 1521–1535.MathSciNetMATHCrossRefGoogle Scholar
  3. [At]
    A. O. L. Atkin, Weierstrass points at cusps of X O (N), Ann. of Math. 85 (1967), 42–45.MathSciNetMATHCrossRefGoogle Scholar
  4. [F-K]
    H. M. Farkas and I. Kra, “Riemann surfaces,” Springer-Verlag, New York, 1992.Google Scholar
  5. [K-Z]
    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.Google Scholar
  6. [L-N]
    J. Lehner and M. Newman, Weierstrass points on Γ0 (N), Ann. of Math. 79 (1964), 360–368.MathSciNetMATHCrossRefGoogle Scholar
  7. [O1]
    A. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462.MathSciNetMATHGoogle Scholar
  8. [O2]
    A. Ogg, On the Weierstrass points of Xo(N), Illinois J. Math. 22 (1978), 31–35.MathSciNetMATHGoogle Scholar
  9. [R]
    D. Rohrlich, Weierstrass points and modular forms, Illinois J. Math. 29 (1985), 134–141.MathSciNetMATHGoogle Scholar
  10. [Sw]
    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.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Scott Ahlgren
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations