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.

No abstract

Let π be a fixed Hecke-Maass cusp form for SLp3, Zq and χ be a primitive Dirichlet character modulo M , which we assume to be a prime. Let Lps, π b χq be the L-function associated to π b χ. In this paper, introducing some variants to previous methods, we establish the bound Lp1{2 `it, π b χq ! π,δ pM p|t| `1qq 3{4´δ for any δ ă 1{36.

Abstract In this paper, we introduce a simple Bessel $\delta $-method to the theory of exponential sums for $\textrm{GL}_2$. Some results of Jutila on exponential sums are generalized in a less technical manner to holomorphic newforms of arbitrary level and nebentypus. In particular, this gives a short proof for the Weyl-type subconvex bound in the $t$-aspect for the associated $L$-functions.

Let π be a fixed Hecke-Maass cusp form for SL(3, Z) and χ be a primitive Dirichlet character modulo M, which we assume to be a prime. Let L(s, π ⊗ χ) be the L-function associated to π ⊗ χ . For any given ε > 0, we establish a subconvex bound L(1/2 + it, π ⊗ χ) π,ε (M(|t| + 1)) 3/4−1/36+ε , uniformly in both the Mand t-aspects.