TY - JOUR
T1 - Vanishing of modular forms at infinity
AU - Ahlgren, Scott
AU - Masri, Nadia
AU - Rouse, Jeremy
PY - 2009/4
Y1 - 2009/4
N2 - We give upper bounds for the maximal order of vanishing at ∞ of a modular form or cusp form of weight k on Γ0(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 ∈ S2(Γ0(Np) ) with ord∞(f) larger than the genus of X0(Np). In particular, this gives a (non-geometric) proof of a theorem of Ogg, which asserts that ∞ is not a Weierstrass point on X0(Np) if p N and X0(N) has genus zero.
AB - We give upper bounds for the maximal order of vanishing at ∞ of a modular form or cusp form of weight k on Γ0(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 ∈ S2(Γ0(Np) ) with ord∞(f) larger than the genus of X0(Np). In particular, this gives a (non-geometric) proof of a theorem of Ogg, which asserts that ∞ is not a Weierstrass point on X0(Np) if p N and X0(N) has genus zero.
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U2 - 10.1090/S0002-9939-08-09768-2
DO - 10.1090/S0002-9939-08-09768-2
M3 - Article
AN - SCOPUS:77950538978
SN - 0002-9939
VL - 137
SP - 1205
EP - 1214
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 4
ER -