Density of modular forms with transcendental zeros

Abstract

For an even positive integer k, let M-k,M-Z(SL2(Z)) be the set of modular forms of weight k on SL2(Z) with integral Fourier coefficients. Let M-k,Z(tran) (SL2(Z)) be the subset of M-k,M-Z(SL2(Z)) consisting of modular forms with only transcendental zeros on the upper half plane H except all elliptic points of SL2(Z). For a modular form f(z) = Sigma(infinity)(n=0) a(f)(n)(e2 pi inz) of weight k(f), let omega(f) := Sigma(rk(f))(n=0) |a(f) (n)|, where r(k(f)) = dim(C) M-k(f),M-Z(SL2(Z)) circle times C -1. In this paper, we prove that if k = 12 or k > 16, then #{f is an element of M-k,Z(tran) (SL2(Z)) : omega(f) <= X}/#{f is an element of M-k,M-Z(SL2(Z)) : omega(f) <= X} = 1 - alpha k/X = O(1/X-2) as X ->infinity where alpha(k) denotes the sum of the volumes of certain polytopes. Moreover, if we let M-Z = (Uk =0Mk,Z)-M-infinity(SL2(Z)) (resp. M-Z(tran)= (Uk=0Mk,Ztran)-M-infinity (SL2(Z))) and phi is a monotone increasing function on R+ such that phi(x + 1) - phi(x) >= Cx(2) for some positive number C, then we prove lim(X ->infinity) #{f is an element of M-Z(tran) : omega(f) + phi(k(f)) <= X}/#{f is an element of M-Z : omega(f) + phi(k(f)) <= X} = 1. (C) 2021 Elsevier Inc. All rights reserved.