We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$, whereas the previous best was $T^{1/3}$, from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$. Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $, this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$.

Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

In this paper, we prove that the Sato–Tate conjecture for primitive Maass forms holds on average. We also investigate the rate of convergence in the Sato–Tate conjecture and establish some estimates of the discrepancy with respect to the Sato–Tate measure on the average of primitive Maass forms.

We investigate the large weight ($k\to\infty$) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of $L$-functions, $\{L(s,\phi \times f): f \in H_k\}$ and $\{L(s,\phi \times {\rm sym}^2 f): f \in H_k\}$; here $\phi$ is a fixed even Hecke–Maass cusp form and $H_k$ is a Hecke eigenbasis for the space $H_k$ of holomorphic cusp forms of weight $k$ for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1- and 2-level densities, we find evidence of underlying symplectic and SO(even) symmetry, respectively. This should be contrasted with previous results of Iwaniec–Luo–Sarnak for the families $\{L(s,f): f\in H_k\}$ and $\{L(s,{\rm sym}^2f): f\in H_k\}$, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus, the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,f×g) and L(s, sym2(g)× sym2(g)), where f,g are newforms, holomorphic or of Maass type, on the upper half plane, and sym2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym2(g).