On the Real Zeroes of Half-integral Weight Hecke Cusp Forms, II

2026/01/31 by Jesse Jääsaari
Mathematics · #math.NT

paper · pdf

Some updates and corrections; 46 pages

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We show that for ≫ K² of the half-integral weight Hecke cusp forms in Kohnen plus subspaces with weight bounded by a large parameter K, the number of "real" zeroes grows at the expected rate. A key technical step in the proof is to obtain sharp bounds for the mollified first and second moments of quadratic twists of modular L-functions.

Citations

• On the Real Zeroes of Half-integral Weight Hecke Cusp Forms
• Mass equidistribution for Saito-Kurokawa lifts
• Quantum variance for holomorphic Hecke cusp forms on the vertical\n geodesic
• Weighted central limit theorems for central values of L-functions
• Signs of Fourier coefficients of half-integral weight modular forms
• Large sums of Hecke eigenvalues of holomorphic cusp forms
• Moments of L-functions and Liouville-Green method
• Small scale distribution of zeros and mass of modular forms
• Mass equidistribution for Hecke eigenforms
• On the asymptotic distribution of zeros of modular forms
• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189
• Low lying zeros of families of L-functions

Related

• On the Real Zeroes of Half-integral Weight Hecke Cusp Forms
• On the number of zeros of L-functions attached to cusp forms of half-integral weight
• A zero-density estimate for L-functions associated with \rm GL(3) Hecke--Maass cusp forms
• On the non-vanishing of modular L-values and fourier coefficients of cusp forms
• Signs of Fourier coefficients of half-integral weight modular forms