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Bingrong Huang

Publications and source records attributed to Bingrong Huang.

At least 19 recordsLinked to original sources
arXiv2026

Joint distribution of Hecke eigenforms

In this paper, we formulate conjectures on the joint distribution of several Hecke eigenforms. We prove an asymptotic formula of the joint mass of two Hecke eigenforms under the generalized Riemann Hypothesis (GRH) and the generalized Ramanujan conjecture (GRC). We also show that a higher decorrelation of two Hecke eigenforms asymptotically vanishes under GRH. As a consequence, we prove an asymptotic formula for the first moment of the triple product $L$-functions under GRH and GRC.

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arXiv2026

Mixed moments of Hecke eigenforms and $L$-functions

In this paper, we establish estimates for the expectation and variance of the mixed $(2,2)$-moment of two Hecke eigenforms of distinct weights. Our results yield applications to triple product $L$-functions. The proofs are based on moments of $L$-functions.

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arXiv2025

Quantum variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series

In this paper, we give the upper bounds on the variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series respectively. For the cusp form case, the bound comes from a large sieve inequality for symmetric cubes. We also give some nontrivial bounds for higher moments of symmetric cube $L$-functions. For the Eisenstein series case, the upper bound comes from Lindelöf-on-average type bounds for various $L$-functions. In particular, we establish the sharp upper bounds for the fourth moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ $L$-functions and the eighth moment of $\mathrm{GL}(2)$ $L$-functions around special points $1/2+it_j$. Our proof is based on the work of Chandee and Li \cite{C-L20} about bounding the second moment of $\mathrm{GL}(4)\times \mathrm{GL}(2)$ $L$-functions.

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arXiv2025

On the supremum of random cusp forms

A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor.

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arXiv2024

Effective decorrelation of Hecke eigenforms

In this paper, we prove effective quantitative decorrelation of values of two Hecke eigenforms as the weight goes to infinity. As consequences, we get an effective version of equidistribution of mass and zeros of certain linear combinations of Hecke eigenforms.

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arXiv2024

Joint value distribution of Hecke--Maass forms

In this paper, we formulate a conjecture on joint distribution of Hecke--Maass cusp forms. To support our conjecture, we prove two conditional results on joint moments of two Hecke--Maass cusp forms, which confirms statistical independence of orthogonal cusp forms.

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arXiv2024

Mixed moments of $\rm GL(2)$ and symmetric square $L$-functions

In this paper, we prove asymptotic formulas of mixed moments of $\rm GL(2)$ and its symmetric square $L$-functions for both Hecke--Maass cusp forms and holomorphic Hecke eigenforms in short intervals. As an application, we prove quantitative simultaneous non-vanishing of central values of these $L$-functions.

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arXiv2023

Extreme central $L$-values of almost prime quadratic twists of elliptic curves

In this paper, we prove the extreme values of $L$-functions at the central point for almost prime quadratic twists of an elliptic curve. As an application, we get the extreme values for the Tate--Shafarevich groups in the quadratic twist family of an elliptic curve under the Birth--Swinnerton-Dyer conjecture.

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arXiv2023

On the Rankin--Selberg problem, II

In this paper, we improve our bounds on the Rankin--Selberg problem. That is, we obtain smaller error term of the second moment of Fourier coefficients of a $\rm GL(2)$ cusp form (both holomorphic and Maass).

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arXiv2023

The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions

In this paper, we prove the smooth cubic moments vanish for the Hecke--Maass cusp forms, which gives a new case of the random wave conjecture. In fact, we can prove a polynomial decay for the smooth cubic moments, while for the smooth second moment (i.e. QUE) no rate of decay is known unconditionally for general Hecke--Maass cusp forms. The proof bases on various estimates of moments of central $L$-values. We prove the Lindelöf on average bound for the first moment of $\rm GL(3)\times GL(2)$ $L$-functions in short intervals of the subconvexity strength length, and the convexity strength upper bound for the mixed moment of $\rm GL(2)$ and the triple product $L$-functions. In particular, we prove new subconvexity bounds of certain $\rm GL(3)\times GL(2)$ $L$-functions.

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arXiv2022

Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions

Let $ϕ$ and $ϕ'$ be two $\textrm{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $ϕ=ϕ'\textrm{ or }\widetilde{ϕ'}$ if there exists a nonzero constant $κ$ such that $$L(\frac{1}{2},ϕ\otimes χ_{8d})=κL(\frac{1}{2},ϕ'\otimes χ_{8d})$$ for all positive odd square-free positive $d$. Here $\widetilde{ϕ'}$ is dual form of $ϕ'$ and $χ_{8d}$ is the quadratic character $(\frac{8d}{\cdot})$. To prove this, we obtain asymptotic formulas for twisted first moment of central values of quadratic twisted $L$-functions on $\textrm{GL}(3)$, which will have many other applications.

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arXiv2021

Uniform bounds for $\rm GL(3) \times GL(2)$ $L$-functions

In this paper, we prove uniform bounds for $\rm GL (3)\times GL(2)$ $L$-functions in the $\rm GL(2)$ spectral aspect and the $t$ aspect by a delta method. More precisely, let $ϕ$ be a Hecke--Maass cusp form for $\rm SL(3,\mathbb{Z})$ and $f$ a Hecke--Maass cusp form for $\rm SL(2,\mathbb{Z})$ with the spectral parameter $t_f$. Then for $t\in\mathbb{R}$ and any $\varepsilon>0$, we have \[ L(1/2+it,ϕ\times f) \ll_{ϕ,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \] Moreover, we get subconvexity bounds for $L(1/2+it,ϕ\times f)$ whenever $|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon}$.

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arXiv2021

Analytic twists of $\rm GL_2\times\rm GL_2$ automorphic forms

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $λ_f(n)$ and $λ_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$ \sum_{n=1}^{\infty}λ_f(n) λ_g(n)e\left(t φ\left(\frac{n}{X}\right)\right)V\left(\frac{n}{X}\right), $$ where $e(x)=e^{2πix}$, $V(x)\in \mathcal{C}_c^{\infty}(1,2)$, $t\geq 1$ is a large parameter and $φ(x)$ is some nonlinear real valued smooth function. Applications of these estimates include a subconvex bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ in the $t$-aspect, an improved estimate for a nonlinear exponential twisted sum and the following asymptotic formula for the sum of the Fourier coefficients of certain $\rm{GL}_5$ Eisenstein series $$ \sum_{n \leq X}λ_{1\boxplus(f\times g)}(n) =L(1,f\times g)X + O(X^{\frac{2}{3}-\frac{1}{356}+\varepsilon}) $$ for any $\varepsilon>0$.

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arXiv2021

On the Rankin--Selberg problem

In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.

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arXiv2021

An orthogonality relation for GL(4,R)

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm{GL}(1)$) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm{GL}(2)$ and $\mathrm{GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm{GL}(4,\mathbb{R})$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula. An appendix by Bingrong Huang gives new bounds for the relevant Kloosterman sums.

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arXiv2021

Effective joint equidistribution of primitive rational points on expanding horospheres

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems -- an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical subgroup in any dimension. We apply our result to provide a rate of convergence to the limiting distribution for the appropriately rescaled diameters of random circulant graphs.

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