Proportion of modular forms with transcendental zeros for general levels

초록

Let Gamma be a congruence subgroup such that Gamma(1)(N) subset of Gamma subset of Gamma(0)(N) for some positive integer N. For a positive integer k, let M-k,M-Z(Gamma) be the set of modular forms of weight k on Gamma with integral Fourier coefficients. Let R-k(Gamma) be the set of common zeros in the upper half plane H of all the modular forms of weight k on Gamma. In this note, we prove that the density of modular forms in M-k,M-Z(Gamma) with an algebraic zero z is not an element of R-k(Gamma) is zero.

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