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

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $Γ_0(4)\backslash\mathbb H$ near a cusp at infinity. In analogue of the Ghosh-Sarnak conjecture for classical holomorphic Hecke cusp forms, one expects that almost all of the zeroes sufficiently close to this cusp lie on two vertical geodesics $\Re(s)=-1/2$ and $\Re(s)=0$ as the weight tends to infinity. We show that, for $\gg_\varepsilon K^2/(\log K)^{3/2+\varepsilon}$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large constant $K$, the number of such “real” zeroes grows almost at the expected rate. We also obtain a weaker lower bound for the number of real zeroes that holds for a positive proportion of forms. One of the key ingredients is the asymptotic evaluation of averaged first and second moments of quadratic twists of modular $L$-functions.

💡 Research Summary

The paper investigates the fine‑scale distribution of zeros of half‑integral weight Hecke cusp forms on the modular surface Γ₀(4)\H, focusing on the region near the cusp at infinity. Inspired by the Ghosh–Sarnak conjecture for classical holomorphic Hecke cusp forms, the authors expect that, when the weight k + ½ tends to infinity, almost all zeros lying sufficiently close to the cusp should lie on the two vertical geodesics Re s = −½ and Re s = 0. These are called “real” zeros because the forms are real‑valued on those lines after a natural normalization of Fourier coefficients.

The main results are two quantitative theorems. Let K be a large parameter and consider the family S_K of half‑integral weight Hecke eigenforms of weight k + ½ with k≈K lying in the Kohnen plus subspace. The cardinality of this family is ≍K². For any fixed ε>0 and for each of the two geodesics δ₁ (Re s = −½) and δ₂ (Re s = 0), the authors prove:

1. Theorem 1.1 (almost‑optimal lower bound). For at least ≫_ε K²/(log K)^{3/2+ε} forms in S_K, the number of real zeros in the Siegel set F_Y={z∈F: Im z≥Y} satisfies
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