Zeros of certain combinations of Eisenstein series

TL;DR

The paper proves that for large weights, the zeros of a specific combination of Eisenstein series lie on the boundary of the fundamental domain, and provides formulas for counting these zeros on boundary segments.

Contribution

It establishes the boundary location of zeros for certain Eisenstein series combinations and derives explicit formulas for their zero counts.

Findings

Zeros lie on the boundary for large weights

Formulas for zeros on the bottom arc and sides

Approximation of Eisenstein series using Jacobi theta function

Abstract

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_k(z) E_{\ell}(z) - E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. We moreover find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $x = \pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.