Zeros of modular forms and Faber polynomials (Q6540008)

This article studies the zeros of cusp forms of large weight \(k\) for the modular group, with large order of vanishing at infinity and a fixed number \(D\) of finite zeros in the fundamental domain, as the weight \(k\) tends to infinity. The article shows that the zeros of these forms cluster near \(D\) vertical lines, with the zeros of weight \(k\) form lying at height approximately \(\log k\). This is in contrast to the distributions of zeros of other families of modular forms previously studied: the zeros of Eisenstein series lie on the circular part of the boundary of the fundamental domain, and the zeros of cuspidal Hecke eigenforms are equidistributed in the fundamental domain.\N\NThe proof makes use of Faber polynomials. In the article, it is shown that the Faber polynomials associated to these forms, when suitably renormalised, converge to the truncated exponential polynomial of degree \(D\).