Young's restricted quantum unique ergodicity conjecture for holomorphic cusp forms

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Let ψ ⁣:R+→R\psi\colon \mathbb{R}^+ \to \mathbb{R} be a smooth compactly supported function, and let f(z)f(z) run over L2L^2-normalized holomorphic Hecke cusp forms of weight kk. Young's restricted quantum unique ergodicity conjecture.

lim⁡k→∞∫0∞yk∣f(iy)∣2ψ(y)dyy=3π∫0∞ψ(y)dyy.\lim_{k\to \infty} \int_0^\infty y^{k} |f(iy)|^2 \psi(y) \frac{dy}{y} = \frac{3}{\pi} \int_0^\infty \psi(y)\frac{dy}{y}.

This is a quantum unique ergodicity statement restricted to the vertical geodesic through ii; unlike the corresponding unrestricted QUE results, the conjecture concerns the mass of holomorphic cusp forms along a lower-dimensional geodesic segment and its status is not resolved in the supplied source.

References

Primary source

Qingfeng Sun and Qizhi Zhang, “Mass Distribution for holomorphic cusp forms on the vertical geodesic”, arXiv:2408.15259 (2024).

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