What question did this study set out to answer?

The research aims to identify and characterize the zeros of Hecke polynomials associated with weak eigenforms of weight 2.

May 9, 2026

Zeros of Hecke Polynomials Arising From Weak Eigenforms

Key Points

The research aims to identify and characterize the zeros of Hecke polynomials associated with weak eigenforms of weight 2.
Constructed Hecke polynomials from weak Hecke eigenforms.
Analyzed Hecke orbits on the unit-circle arc using cosine term isolation and tail control via Maass–Poincaré series.
Extended existing theories from holomorphic forms to harmonic Maass forms.
Every zero is simple and lies within the interval [0, 1728].
Established a clean degree–monicity formula for the zeros.
Provided clear criteria for identifying zeros specifically at 0 and 1728.

Abstract

We attach Hecke polynomials Formula: see text(Formula: see text) to weak Hecke eigenforms Formula: see text of weight 2 – Formula: see text and show that, for large Formula: see text, every zero is simple and lies in 0, 1728. The construction pulls back a weakly holomorphic Hecke combination of Formula: see text along Formula: see text; the analysis follows Hecke orbits on the unit-circle arc Formula: see text, isolating a dominant “cosine” term and controlling the tail via Maass–Poincaré series and Whittaker/Bessel bounds. This extends the Rankin–Swinnerton-Dyer/Asai–Kaneko–Ninomiya picture from holomorphic forms to a broad class of harmonic Maass forms and yields a clean degree–monicity formula and simple criteria for zeros at 0 and 1728.

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