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.
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Kilger, K. Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms. Ramanujan J 17, 321–330 (2008). https://doi.org/10.1007/s11139-007-9018-8
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DOI: https://doi.org/10.1007/s11139-007-9018-8