The variation of zeros of the Miller basis

Abstract

We exhibit a connection between the variation of zeros in the Miller basis of modular forms $q^m+O(q^{\ell +1})$ and a logarithmic version ${\mathcal{S}}_{\delta }$ of the Szegő curve, where $\delta =m/\ell$. When $\delta <0.6194$ we show that all the zeros are on the unit arc for $k\gg 0$, while if $\delta$ is asymptotically close to 1, we show that all the zeros lie on ${\mathcal{S}}_{\delta }$. In general, we posit that for all $\delta$, the zeros are located on the union of the unit arc and the log Szegő curve, obtaining a partial result, and find conjectural thresholds for $m/\ell$ with all zeros on the unit arc, and no zeros on the arc. Finally, we enumerate all algebraic zeros of Miller forms up to $\ell -m\le 25$.