Related papers: Spectral Moment Formulae for $GL(3)\times GL(2)$ $…

This work is the second in a series, following Part I (Algebra Number Theory 18.10 (2024)) and preceding Part III (Math. Ann. 391.1 (2025)). We continue our investigation of spectral moments of $\hbox{GL}(3)\times \hbox{GL}(2)$…

Spectral moment formulae of various shapes have proven to be very successful in studying the statistics of central $L$-values. In this article, we establish, in a completely explicit fashion, such formulae for the family of $GL(3)\times…

We give an asymptotic formula with power saving error term for the twisted first moment of symmetric square L-functions on GL(3) in the level aspect. As applications, we obtain non-vanishing results as well as lower bounds of the expected…

We compute an asymptotic formula for the twisted moment of GL(3)xGL(2) L-functions and their derivatives. As an application we prove that symmetric-square lifts of GL(2) Maass forms are uniquely determined by the central values of the…

A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke…

A formula connecting a moment of L-functions and a dual moment in a way that interchanges the roles of certain key parameters on both sides is known as a reciprocity relation. We establish a reciprocity relation for a first moment of GL(2)…

Let $F$ be a Hecke-Maa\ss\ cusp form for $\mathrm{SL}(3,\mathbb{Z})$. We obtain the first non-trivial upper bound of the second moment of $L(F,s)$ in $t$-aspect: $$\int_{T}^{2T}|L(F,1/2+it)|^2 dt\ll_{F,\varepsilon}…

In this paper, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms for $\rm SL(3,\mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions…

We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of $L$-functions for $\mathrm{GL}_1$ with the third moment of $L$-functions for $\mathrm{GL}_2$ over number fields, studied…

We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking $\mathrm{GL}_3$ and $\mathrm{GL}_2$ automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal…

We prove a new case of spectral reciprocity formulae for the product of $GL(n+1) \times GL(n)$ and $GL(n) \times GL(n-1)$ Rankin-Selberg $L$-functions ($n \geq 3$), which are first developed by Blomer and Khan in \cite{BK17} for degree 8…

In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we…

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…

For a fixed SL(3, Z) Maass form g, we consider the family of L-functions L(g \times u_j, s) where u_j runs over the family of Hecke-Maass cusp forms on SL(2,Z). We obtain an estimate for the second moment of this family of L-functions at…

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $\mu$ in generic position i.e. away from Weyl chamber walls and…

In this paper, we study the second moment for $GL(2)\times GL(2)$ $L$-functions $L(\frac{1}{2},f\times g)$, which leads to a uniform subconvexity bound in the spectral aspect. In particular, if either $f$ or $g$ is a dihedral Maass newform,…

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the…

We compute the second moment of a certain family of Rankin-Selberg $L$-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n). Our bound is as strong as the Lindel\"of hypothesis on average, and recovers individually the…

In this paper, we prove the conjectured order lower bound for the $k$-th moment of central values of quadratic twisted self-dual $\textrm{GL}(3)$ $L$-functions for all $k\geq 1$, based on our recent work on the twisted first moment of…

We employ a regularized relative trace formula to establish a second moment estimate for twisted $L$-functions across all aspects over a number field. Our results yield hybrid subconvex bounds for both Hecke $L$-functions and twisted…