Zeros of modular forms and Faber polynomials

Abstract

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number $$ of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near $$ vertical lines, with the zeros of a weight $$ form lying at height approximately $$. This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree $$.