Notes on Weierstrass Points of Modular Curves $X_0(N)$

Abstract

We give conditions when the fixed points by the partial Atkin-Lehner involutions on X0(N ) are Weierstrass points as an extension of the result by Lehner and Newman [16]. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on X0(N ) are Weierstrass points or not.

Weierstrass points on modular curves X0(N) fixed by the Atkin–Lehner involutions

“…Theorem 3.4. [12] The points fixed by W N for N ≤ 50 are Weierstrass points on X 0 (N) with g 0 (N) > 1 except possibly for the following values First, only a finite number of Weierstrass points can exist on X 0 (N), and if g 0 (N) ≤ 1, then are no such points at all. So we have the following theorem: Lewittes [17] proved that if X 0 (N) is a hyperelliptic modular curve, then any involution on X 0 (N) either has no fixed points or has only non Weierstrass fixed points or is the hyperelliptic involution.…”

“…In addition, Jeon [10,11] has computed all Weierstrass points on the hyperelliptic curves X 1 (N) and X 0 (N). Recently Im, Jeon and Kim [12] have generalised the result of Lehner and Newman [1] by giving conditions when the points fixed by the partial Atkin-Lehner involution on X 0 (N) are Weierstrass points and have determined whether the points fixed by the full Atkin-Lehner involution on X 0 (N) are Weierstrass points or not. In this paper, we have determined which of the points fixed by W Q on X 0 (N) are Weierstrass points and found Weierstrass points on modular curves X 0 (N) for N ≤ 50 fixed by the partial and the full Atkin-Lehner involutions.…”

Weierstrass points on and supersingular j-invariants

“…To compute the right-hand side, we begin with the following. (8,5), (8,11) (mod 12), (6,7), (10,7), (6,11), (10, 11) (mod 12),…”

“…Specifically, the points of X 0 (N ) parameterize isomorphism classes of pairs (E, C) where E is an elliptic curve over C and C is a cyclic subgroup of E of order N . Weierstrass points on X 0 (N ) have been studied by a number of authors (see, for example, [3][4][5][6]12,13,15,17,20,22,23], and [10]). An interesting open question is to determine those N for which the cusp ∞ is a Weierstrass point.…”

“…Then since x and (x − 1728) are coprime to S p (x), we have 10) where the two quotients reduce to polynomials. Now on the left of (6.10), we write S…”