Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Abstract

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X 0 (N ) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X 0 (N ), if N = pM with p prime, p M and the genus of X 0 (M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin-Lehner involution. This gives, in a sense, a generalization of Ogg's Theorem in some cases. Statement of problemLet H be the complex upper-half plane andThen 0 (N ) acts on H by fractional linear transformations. Let X 0 (N ) be the compact Riemann surface, i.e. the algebraic curve obtained from 0 (N )\H by adding the cusps and g(N) the genus of X 0 (N ). These curves play a fundamental role in arithmetic, as each X 0 (N ) is the moduli space of elliptic curves with prescribed cyclic subgroup of order N . A point P ∈ X 0 (N ), by definition, is a Weierstrass point, if there exists a nonzero function on K. Kilger ( ) Mathematisches

Vanishing of modular forms at infinity

“…For example, in the space M 6 (Γ 0 (35)), there is a form whose q-expansion begins with q 21 + · · · . On the other hand, we have The proofs of these results use techniques similar to those in [8], [7]. In order to prove these theorems, we will establish the analogous results in characteristic p. In particular, we give an improvement of a well-known theorem of Sturm on the maximal order of vanishing of a modular form in characteristic p. The tools involve facts on the integrality of modular forms, a modification of Sturm's original result to account for forced vanishing at the elliptic points, the trace map, the theory of modular forms mod p, and a recent result of Kilbourn [6], which, extending results of Deligne-Rapoport [3] and Weissauer [14], gives bounds for the p-adic valuation of the image of a cusp form f ∈ S k (Γ 0 (N )) under the Atkin-Lehner operator W …”

“…Recently, Kohnen [8] and Kilger [7] have used techniques from the theory of modular forms mod p to reprove Ogg's result for certain curves X 0 (p ) when p and are distinct primes. As a corollary to our first theorem, we obtain a proof of Ogg's result which uses only standard facts from the theory of modular forms mod p.…”

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

“…For the purpose, we refer to a paper by Klaise [12] in which all the orders of class number 2 and 3 are determined and an algorithm to find all orders of class number up to 100 is suggested. Let 6,8,9,10,12,13,16,18,22,25,28, 37, 58}, 19,23,27,31,43, 67, 163}, 17,20,21,24,30,32,33,34,36,39,40,42,45,46,48,49,52,55,57,60,63,64,70,72,73,78,82,85,88,93,97,100,102,112,130,133,142,148,177,190,…”

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

“…The Weierstrass points on modular curves have been studied by Lehner and Newman in [1]; they have given conditions when the cusp at infinity is a Weierstrass point on X 0 (N) for N 5 4n, 9n, and Atkin [2] has given conditions for the case of N 5 p 2 n where p is a prime ≥ 5. Besides, Ogg [3], Kohnen [4,5] and Kilger [6] have given some conditions when the cusp at infinity is not a Weierstrass point on X 0 (N) for certain N. Also, Ono [7] and Rohrlich [8] have studied Weierstrass points on X 0 (p) for some primes p. And Choi [9] has shown that the cusp 1 2 is a Weierstrass point of Γ 1 (4p) when p is a prime > 7. In addition, Jeon [10,11] has computed all Weierstrass points on the hyperelliptic curves X 1 (N) and X 0 (N).…”