Voronoï summation via switching cusps

We consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form of general level N at the various cusps of $$\Gamma _{0}(N)\backslash \mathbb {H}$$ Γ 0 ( N ) \ H . We explain how to compute these coefficients via the local theory of p-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin–Lehner relations.

Analytic Twists of GL3 × GL2 Automorphic Forms

Let $\pi $ be a Hecke–Maass cusp form for $\textrm{SL}_3(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _{\pi }(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\textrm{SL}_2(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum $$\begin{align*}& \sum_{r,n\geq 1}\lambda_{\pi}(n,r)\lambda_f(n)e\left(t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right), \end{align*}$$where $e(x)=e^{2\pi ix}$, $V(x)\in \mathcal{C}_c^{\infty }(0,\infty )$, $t\geq 1$ is a large parameter and $\varphi (x)$ is some real-valued smooth function. As applications, we give an improved subconvexity bound for $\textrm{GL}_3\times \textrm{GL}_2$ $L$-functions in the $t$-aspect and under the Ramanujan--Petersson conjecture we derive the following bound for sums of $\textrm{GL}_3\times \textrm{GL}_2$ Fourier coefficients $$\begin{align*}& \sum_{r^2n\leq x}\lambda_{\pi}(r,n)\lambda_f(n)\ll_{\pi, f, \varepsilon} x^{5/7-1/364+\varepsilon} \end{align*}$$for any $\varepsilon>0$, which breaks for the 1st time the barrier $O(x^{5/7+\varepsilon })$ in a work by Friedlander–Iwaniec.

Bounds for GL3L-functions in depth aspect

Let f be a Hecke–Maass cusp form for {\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor {\mathfrak{q}=p^{\kappa}}, with p prime. We prove the subconvexity boundL\Big{(}\frac{1}{2},\pi\otimes\chi\Big{)}\ll_{p,\pi,\varepsilon}\mathfrak{q}^{% 3/4-3/40+\varepsilon}for any {\varepsilon>0}, where the dependence of the implied constant on p is explicit and polynomial.

Rapid computation of L-functions attached to Maass forms

Let [Formula: see text] be a degree-[Formula: see text] [Formula: see text]-function associated to a Maass cusp form. We explore an algorithm that evaluates [Formula: see text] values of [Formula: see text] on the critical line in time [Formula: see text]. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution.

Non-vanishing of the first derivative of GL(3) ×GL(2) L-functions

Let [Formula: see text] be a fixed self-dual Hecke–Maass cusp form for [Formula: see text] and [Formula: see text] be an orthogonal basis of odd Hecke–Maass cusp forms for [Formula: see text]. We prove an asymptotic formula for the average of the first derivative of the Rankin–Selberg [Formula: see text]-function of [Formula: see text] and [Formula: see text] at the center point [Formula: see text]. This implies the non-vanishing results for the first derivative of these [Formula: see text]-functions at the center point [Formula: see text].

On sums of Fourier coefficients of Maass cusp forms

Let [Formula: see text] be a Hecke–Maass cusp form, and [Formula: see text] be its [Formula: see text]th Fourier coefficient at the cusp infinity. In this paper, we are interested in the estimation on sums [Formula: see text] for [Formula: see text] We are able to improve previous results by introducing some inequalities concerning Fourier coefficients and other techniques.

Power moments of automorphic L-function attached to Maass forms

Let [Formula: see text] be a normalized Maass cusp form for [Formula: see text]. For [Formula: see text], we define [Formula: see text] [Formula: see text] as the supremum of all numbers [Formula: see text] such that [Formula: see text] where [Formula: see text] is the automorphic [Formula: see text]-function attached to [Formula: see text]. In this paper, we shall establish the lower bounds of [Formula: see text] for [Formula: see text] and obtain asymptotic formulas for the second, fourth and sixth powers of [Formula: see text].

On effective determination of Maass forms from central values of Rankin–Selberg L-function

We address the problem of identifying a Hecke–Maass cusp form

Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms

Let$\nu _{f}(n)$be the$n\mathrm{th}$normalized Fourier coefficient of a Hecke–Maass cusp form$f$for${\rm SL }(2,\mathbb{Z})$and let$\alpha $be a real number. We prove strong oscillations of the argument of$\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$as$n$takes consecutive integral values.