On the Asymptotic Distribution of Zeros of Modular Forms

1.1. Our purpose in this note is to study the limiting distribution of zeros of modular forms. We review some definitions: A modular form of weight k for SL2(Z) is a holomorphic function on the upper half-plane H, transforming as f( cz+d ) = (cz + d)f(z), for all ( a b c d ) ∈ SL2(Z) (this forces k to be even), and “holomorphic at the cusp” (see § 2.1). A form is cuspidal if it vanishes at the cusp. For a modular form of weight k, let ν(f) be the number of inequivalent zeros of f in H, with the convention that a zero at z is counted with weight w(z) inverse to the number of elements of SL2(Z)/{±I} fixing z. Then ν(f) ≤ k/12. For a sequence of modular forms, where we assume that the number of inequivalent zeros tends to infinity, we would like to examine the manner in which the resulting configuration of zeros is distributed in the modular domain SL2(Z)\H.