In this paper, we investigate the mean square estimate for the logarithmic derivative of the Godement--Jacquet $L$-function $L_f(s)$ assuming the Riemann hypothesis for $L_f(s)$ and Rudnick--Sarnak conjecture.
We observe how certain distribution on $\mathrm{GL}(n)$ that generates Godement-Jacquet $γ$-factors appears naturally in the C*-algebraic normalization process for standard intertwining integrals on $\mathrm{SL}(n+1)$.
Given an induced representation of Langlands type $(π,V)$ of $\mathrm{GL}_n(F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $β$ and Schwartz-Bruhat function $Φ$ for which the Godement-Jacquet zeta integral $Z(s,β,Φ)$ attains the $L$-function $L(s,π)$.
Suppose $F$ is a non-archimedean local field. The classical Godement-Jacquet theory is that one can use Schwartz-Bruhat functions on $n \times n$ matrices $M_n(F)$ to define the local standard $L$-functions on $\mathrm{GL}_n$. The purpose of this partly expository note is to give evidence that there is an analogous and useful "approximate" Godement-Jacquet theory for the standard $L$-functions on the special orthogonal groups $\mathrm{SO}(V)$: One replaces $\mathrm{GL}_n(F)$ with $\mathrm{GSpin}(V)(F)$ and $M_n(F)$ with $\mathrm{Clif}(V)(F)$, the Clifford algebra of $V$. More precisely, we exp…
We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.
We establish Connes's local trace formula (related to the explicit formulae of number theory) for the quaternions. This is done as an application of a study of the central operator H = log(|x|) + log(|y|) in the context of invariant harmonic analysis. The multiplicative analysis of the additive Fourier transform gives a spectral interpretation to generalized ``Tate Gamma functions'' (closely akin to the Godement-Jacquet ``γ(s,π,ψ)'' functions.) The analysis of H leads furthermore to a spectral interpretation for the logarithmic derivatives of these Gamma functions (which are involved in ``expl…
In this paper, we prove a kernel formula of Bessel functions attached to irreducible smooth supercuspidal representations of p-adic $GL(n)$. We also show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell. As an application we give a proof of the local converse conjecture of Jacquet.
The Local Converse Problem is to determine how the family of the local gamma factors $γ(s,π\timesτ,ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, w…
Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain $L$-function. As a consequence we deduce certain local results about the existence of $GL_2(k)$-invariant linear forms on irreducible, admissible representations of $GL_2({\Bbb K})$ for ${\Bbb K}$ a commutative semi-simple cubic algebra over a non-archimedean local field $k$ in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earl…
We prove an analogue of Jacquet's conjecture on the local converse theorem for \ell-adic families of co-Whittaker representations of GL_n(F), where F is a finite extension of Q_p and \ell does not equal p. We also prove an analogue of Jacquet's conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an \ell-adic family. These two theorems are closely related to the local Langlands correspondence in \ell-adic families.
In this note, we prove in full generality a conjecture of Jacquet concerning the nonvanishing of the triple product L-function at the central point. Let $\kay$ be a number field and let $π_i$, $i=1$, 2, 3 be cuspidal automorphic representations of $GL_2(\A)$ such that the product of their central characters is trivial. Then the central value $L(\frac12,π_1\otimesπ_2\otimesπ_3)$ of the triple product L--function is nonzero if and only if there exists a quaternion algebra $B$ over $\kay$ and automorphic forms $f_i^B\in π_i^B$, such that the integral of the product $f_1^B f_2^B f_3^B$ over the di…
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_n\times U_{n+1}$ in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called $*$-generic, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods. The other, built upon Zeta integrals, is expressed in terms of fu…
We study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet-Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuil's locally analytic socle conjecture for $\mathrm{GL}_n(\mathbb{Q}_p)$.
Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be wri…
For a totally real number field $F$ and its adèle ring $\mathbb{A}_F$, let $π$ vary in the set of irreducible cuspidal automorphic representations of ${\rm PGL}_2(\mathbb{A}_F)$ corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power $L$-functions $L(s,{\rm Sym}^r(π))$ weighted by special values of symmetric square $L$-functions $L(\frac{z+1}{2},{\rm Sym}^2(π))$ at $z \in [0, 1]$ in the level aspect. If $0 < z \le 1$, our weighted density in the level aspect has the same symmetry…
