Mass equidistribution for Poincaré series of large index

TL;DR

The paper proves mass equidistribution for Poincaré series $P_{k,m}$ of large index on the modular surface $X=\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}$ in the regime $k\ll m\ll k^{3/2-\varepsilon}$ with $m(k)/k\to\infty$, by combining unfolding with spectral analysis and uniform bounds on Fourier coefficients. The authors develop detailed bounds for the Fourier coefficients $p_k(m;n)$ and analyze the unfolding integral against both Maass cusp forms and Eisenstein series, establishing decay of nontrivial spectral contributions and confirming mass equidistribution with respect to the hyperbolic measure; they also derive a diagonal normalization $\langle P_{k,m},P_{k,m}\rangle \sim \dfrac{\Gamma(k-1)}{(4\pi m)^{k-1}}$. As a consequence, the zeros of $P_{k,m}$ are shown to be uniformly distributed in $X$ within the same index regime, connecting mass distribution to zero statistics in this holomorphic setting. The results extend holomorphic QUE-type phenomena to a natural non-Hecke family and suggest potential generalizations to broader growth ranges for $m$.

Abstract

Let $P_{k,m}$ denote the Poincaré series of weight $k$ and index $m$ for the full modular group $\mathrm{SL}_2(\mathbb{Z})$, and let $\{P_{k,m}\}$ be a sequence of Poincaré series for which $m(k)$ satisfies $m(k) / k \rightarrow\infty$ and $m(k) \ll k^{\frac{3}{2} - ε}$. We prove that the $L^2$ mass of such a sequence equidistributes on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure as $k$ goes to infinity. As a consequence, we deduce that the zeros of such a sequence $\{P_{k,m}\}$ become uniformly distributed in $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure.

Theorems & Definitions (23)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Proposition 2.2
• Remark
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5