Existence and Weyl's law for spherical cusp forms

Let $F$ be a number field and $n\geqslant 1$ an integer. The universal family is the set $\mathfrak{F}$ of all unitary cuspidal automorphic representations on ${\rm GL}_n$ over $F$, ordered by their analytic conductor. We prove an asymptotic for the size of the truncated universal family $\mathfrak{F}(Q)$ as $Q\rightarrow\infty$, under a spherical assumption at the archimedean places when $n\geqslant 3$. We interpret the leading term constant geometrically and conjecturally d...

Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.

We study the number $N_{\mathrm{sd}}^K(\lambda)$ of self-dual cuspidal automorphic representations of $GL_N(\mathbb{A_Q})$ which are $K$-spherical with respect to a fixed compact subgroup $K$ and whose Laplacian eigenvalue is $\leq \lambda$. We prove Weak Weyl's Law for $N_{\mathrm{sd}}^K(\lambda)$ in the form that there are positive constants $c_1, c_2$ (depending on $K$) and $d$ such that $c_1\lambda^{d/2}\leq N_{\mathrm{sd}}^K(\lambda)\leq c_2\lambda^{d/2}$ for all suffici...

Let $G$ be a connected and simply connected semisimple algebraic group over $\Bbb Q$ and let $\Gamma\subset G(\Bbb Q)$ be an arithmetic subgroup. Let $K_\infty\subset G(\Bbb R)$ be a maximal compact subgroup and let $d$ be the dimension of the symmetric space $G({\mathbb R})/K_\infty$. Let $\sigma$ be an irreducible unitary representation of $K_\infty$. We prove that for every $\Gamma$ there exists a normal subgroup $\Gamma_1\subset \Gamma$ of finite index such that the quoti...

We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series of representations of G/H coincides with the space of cusp forms.

Let E_lambda be a Hilbert space, whose elements are functions spanned by the eigenfunctions of the Laplace-Beltrami operator associated with an eigenvalue lambda>0. The norm of elements in this space is given by the Petersson inner product. In this paper, the trace of Hecke operators T_n acting on the space E_lambda is computed for congruence subgroups of Gamma_0(N) of square free level, which may be considered as the analogue of the Eichler-Selberg trace formula [11] for non...

In this mostly expository note, we prove explicit formulas for the traces of Hecke operators on spaces of cusp forms fixed by Atkin-Lehner involutions, which are suitable for efficient implementation. In addition, we correct a couple of errors in previously published formulas.

The purpose of this paper is to collect and make explicit the results of Gel'fand, Graev and Piatetski-Shapiro and Miyazaki for the $GL(3)$ cusp forms which are non-trivial on $SO(3,\mathbb{R})$. We give new descriptions of the spaces of cusp forms of minimal $K$-type and from the Fourier-Whittaker expansions of such forms give a complete and completely explicit spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash PSL(3,\mathbb{R}))$, accounting for multiplicities, in the s...

In this paper, we prove a general simple relative trace formula. As an application, we prove a relative analogue of the Weyl law.

We generalize the work of Lindenstrauss and Venkatesh establishing Weyl's Law for cusp forms from the spherical spectrum to arbitrary Archimedean type. Weyl's law for the spherical spectrum gives an asymptotic formula for the number of cusp forms that are bi-spherical in terms of eigenvalue T of the Laplacian. We prove an analogous asymptotic holds for cusp forms with Archimedean type {\tau}, where the main term is multiplied by dim {\tau}. While in the spherical case the sur...