The variation of zeros of the Miller basis

TL;DR

This paper explores the distribution of zeros of Miller basis modular forms, linking their variation to the Szegő curve and establishing conditions for zeros to lie on specific curves.

Contribution

It introduces a connection between zeros of Miller basis forms and the logarithmic Szegő curve, providing new thresholds and partial results on zero locations.

Findings

Zeros lie on the unit arc for δ<0.6194 when k is large

Zeros are on the log Szegő curve when δ is close to 1

All algebraic zeros up to a certain degree are enumerated

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 Szeg\H{o} 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 Szeg\H{o} 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 $\ell-m\leq 25$.