Central values of GL(2) ×GL(3) Rankin–Selberg L-functions

Let [Formula: see text] be a normalized holomorphic cusp form for [Formula: see text] of weight [Formula: see text] with [Formula: see text]. By the Kuznetsov trace formula for [Formula: see text], we obtain the twisted first moment of the central values of [Formula: see text], where [Formula: see text] varies over Hecke–Maass cusp forms for [Formula: see text]. As an application, we show that such [Formula: see text] is determined by [Formula: see text] as [Formula: see text] varies.

A SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].

Plancherel Distribution of Satake Parameters of Maass Cusp Forms on GL3

We prove an equidistribution result for the Satake parameters of Maass cusp forms on $\operatorname{GL}_{3}$ with respect to the p-adic Plancherel measure by using an application of the Kuznetsov trace formula. The techniques developed in this paper deal with the removal of arithmetic weight $L(1,F,\operatorname{Ad})^{-1}$ in the Kuznetsov trace formula on $\operatorname{GL}_{3}$.

BEYOND ENDOSCOPY FOR THE RELATIVE TRACE FORMULA II: GLOBAL THEORY

For the group $G=\operatorname{PGL}_{2}$ we perform a comparison between two relative trace formulas: on the one hand, the relative trace formula of Jacquet for the quotient $T\backslash G/T$, where $T$ is a nontrivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to nonstandard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ ‘Beyond Endoscopy’ program.

An orthogonality relation for a thin family of GL(3) Maass forms

We prove an orthogonality relation for the Fourier–Whittaker coefficients of a thin family of GL(3) Maass forms containing all self-dual forms. This is obtained by analyzing the Kuznetsov trace formula on GL(3) for a certain family of test functions. The method also yields Weyl's law for the same family of Maass forms.