Hecke polynomials for the mock modular form arising from the Delta-function

TL;DR

This paper studies a mock modular form related to Ramanujan's Delta-function, analyzing the zeros of associated Hecke polynomials and showing their distribution properties.

Contribution

It introduces a new mock modular form from the Delta-function and characterizes the zeros of its Hecke polynomials, including their distribution and location.

Findings

Zeros of the Hecke polynomials are distinct and within [0, 1728]

Zeros include special points 0 and 1728

Zeros become equidistributed as m increases

Abstract

We consider a mock modular form $M_{\Delta}(\tau)$ that arises naturally from Ramanujan's Delta-function. It is a weight $-10$ harmonic Maass form whose nonholomorphic part is the "period integral function'' of $\Delta(\tau)$. The Hecke operator $T_{-10}(m)$ acts on this mock modular form in terms of Ramanujan's $\tau(m)$ and a monic degree $m$ polynomial $F_m(x),$ evaluated at $x=j(\tau).$ In analogy with results by Asai, Kaneko, and Ninomiya on the zeros of Hecke polynomials for the $j$-function, we prove that the zeros of each $F_m(x)$, including $x=0$ and $x=1728,$ are distinct and lie in $[0, 1728]$. Additionally, as $m \to +\infty,$ these zeros become equidistributed in $[0, 1728].$