A Short Remark on Weierstrass Points at Infinity on X0(N)

Kohnen, Winfried

doi:10.1007/s00605-003-0054-1

Cited by 14 publications

(12 citation statements)

References 2 publications

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“…The key ingredient in the proof of Theorem 3 is the following proposition, whose proof was inspired by [8], Theorem 1. N = p (p, different primes) such that one of the following conditions hold:…”

Section: Proof Of the Resultsmentioning

confidence: 99%

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

Kilger

2007

Ramanujan J

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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

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Section: Proof Of the Resultsmentioning

confidence: 99%

“…Theorem 1 (Kohnen [8]) Let p, be prime, p = and p ≡ 11 mod 12. Then ∞ is not a Weierstrass point on X 0 (p ) if g( ) = 0.…”

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confidence: 99%

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

Kilger

2007

Ramanujan J

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“…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 …”

Section: The Following Question Is Very Naturalmentioning

confidence: 99%

“…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.…”

Section: The Following Question Is Very Naturalmentioning

confidence: 99%

Vanishing of modular forms at infinity

Ahlgren

Masri

Rouse

2008

Proc. Amer. Math. Soc.

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Abstract. We give upper bounds for the maximal order of vanishing at ∞ of a modular form or cusp form of weight k on Γ 0 (Np) when p N is prime. The results improve the upper bound given by the classical valence formula and the bound (in characteristic p) given by a theorem of Sturm. In many cases the bounds are sharp. As a corollary, we obtain a necessary condition for the existence of a non-zero form f ∈ S 2 (Γ 0 (Np)) with ord ∞ (f ) larger than the genus of X 0 (Np). In particular, this gives a (non-geometric) proof of a theorem of Ogg, which asserts that ∞ is not a Weierstrass point on X 0 (Np) if p N and X 0 (N ) has genus zero.

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“…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.…”

Section: Introductionmentioning

confidence: 99%