Related papers: Higher weight on GL(3), I: The Eisenstein series

In this work we give an explicit formula for the Fourier coefficients of Eisenstein series corresponding to certain arithmetic lattices acting on hyperbolic n+1-space. As a consequence we obtain results on location of all poles of these…

In previous papers it has been shown that the coefficients of terms in the large-$N$ expansion of a certain integrated four-point correlator of superconformal primary operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory are…

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the…

The $SL(3)$ Kuznetsov formula exists in several versions, and has been employed with some success to study automorphic forms on $SL(3)$. In each version, the weight functions on the geometric side are given by multiple integrals with…

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical…

In $2003$, Pei and Wang introduced higher level analogs of the classical Cohen--Eisenstein series. In recent joint work with Beckwith, we found a weight $\frac{1}{2}$ sesquiharmonic preimage of their weight $\frac{3}{2}$ Eisenstein series…

We express the contribution of certain maximal parabolic Eisenstein series to the spectral side of the Arthur--Selberg trace formula for GL$(n)$ in terms of zeroes of Rankin--Selberg $L$-functions, generalizing previous results for GL(2)…

For the p-adic group G=SL (2) , we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are…

In this paper, we give an explicit formula of the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we…

Let $n$ and $k$ be positive integers such that $n$ is even. We derive new global integrals for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan, following a strategy and…

We relate Iwahori-Whittaker functions on metaplectic covers to certain Demazure-Lusztig operators, the latter of which are built from a Weyl group action previously considered by G. Chinta and P. Gunnells. Using a certain combinatorial…

One of the main objectives of the current paper is to revisit the well known Laurent series expansions of the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$ and Dirichlet $L$-function $L(s,\chi)$ at $s=1$. Moreover, we…

Chapter $7$ of Langlands' monograph "On the functional equations satisfied by Eisenstein series" employs a sophisticated residue scheme to construct a portion of the discrete automorphic spectrum. We show, by examples, applications, and…

We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the $n$-fold cover of the split odd orthogonal group $SO_{2r+1}$. If the degree of the cover is odd, then Beineke, Brubaker and Frechette have conjectured…

With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half-integral weight is lopsided; the analytic theory is strong whereas the algebraic lags behind. In this paper, we capitalise on this to establish the…

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…

Spectral moment formulae of various shapes have proven to be very successful in studying the statistics of central $L$-values. In this article, we establish, in a completely explicit fashion, such formulae for the family of $GL(3)\times…

We study a spectral reciprocity formula relating $\mathrm{GL}_3 \times \mathrm{GL}_2$ with $\mathrm{GL}_3 \times \mathrm{GL}_1$ and $\mathrm{GL}_1$ moments of $L$-functions discovered by Kwan. Globally we give an adelic and distributional…

We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of $\mathrm{SL}(2,\mathbb{R})$ due to Bernstein and Reznikov to study triple products of Eisenstein…

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights…