Let E/F be a quadratic extension of number fields. For a cuspidal representation $π$ of SL(2,A_E), we study the non-vanishing of the period integral on SL(2,F)\SL(2,A_F). We characterise the non-vanishing of the period integral of $π$ in terms of $π$ being generic with respect to characters of E\A_E which are trivial on A_F. We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of SL(2,A_E) whose period integral vanishes identically while each local constituent admits an SL(2)-invariant linear functional. Finally, we construct an automorphic representation $π$ on SL(2,A_E) which is abstractly SL(2,A_F) distinguished but none of the elements in the global L-packet determined by $π$ is distinguished by SL(2,A_F).
Let $π_1,π_2$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a non-archimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin-Selberg $L$-factor $L(X,π_1,π_2)$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $π_1$ and $π_2$ such that the local Rankin-Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_F^n$ is equal to $L(X,π_1,π_2)$. We give an initial application of the test vectors to reduction modulo $\ell$ of $\…
This paper proves the existence of cuspidal automorphic forms for a reductive group, invariant under an automorphism of finite order. The techniques used are a local analysis of orbital integrals and the Arthur-Selberg trace formula.
We show the existence of integral models for cuspidal representations of GL(2,q), whose reduction modulo p can be identified with the cokernel of a differential operator on F_{q}[X,Y] defined by J-P. Serre. These integral models come from the crystalline cohomology of the projective curve XY^{q}-X^{q}Y-Z^{q+1}=0. As an application, we can extend a construction of C. Khare and B. Edixhoven (2003) giving a cohomological analogue of the Hasse invariant operator acting on spaces of modp modular forms for GL(2).
Let $F$ be a non-Archimedean locally compact field of residue characteristic $p$, $G$ be an inner form of $GL_n(F)$, $n\ge1$, and $\ell$ be a prime number different from $p$. We give a numerical criterion for an integral $\ell$-adic irreducible cuspidal representation $\tildeρ$ of $G$ to have a super\-cuspidal irreducible reduction mod $\ell$, by counting inertial classes of cuspidal representations that are congruent to the inertial class of $\tildeρ$, generalizing results by Vign{é}ras and Dat. In the case the reduction mod $\ell$ of $\tildeρ$ is not super\-cuspidal irreducible, we show that…
Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $$\int_{Z_{A}GL_{2}(\mathbb{F})\backslash GL_{2}(\mathbb{A}_{\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and oth…
We give constraints on the existence of $(χ,b)$-factors in the global $A$-parameter of a genuine cuspidal automorphic representation $σ$ of a metaplectic group in terms of the invariant, lowest occurrence index, of theta lifts to odd orthogonal groups. We also give a refined result that relates the invariant, first occurrence index, to non-vanishing of period integrals of residues of Eisenstein series associated to the cuspidal datum $χ\otimesσ$. This complements our previous results for symplectic groups.
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to th…
In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $σ$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,σ\timesχ)$ for some character $χ$ of $F^\times\backslash\mathbb{A}^\times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(χ,b)$ in the global Arthur parameter $ψ$ attached to $σ$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein…
Based on Furusawa's theory, we present an integral representation for the L-function L(s,π\times τ), where πis a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where τis an arbitrary cuspidal automorphic representation on GL(2). As an application, a special value result for this L-function in the spirit of Deligne's conjecture is proved.
The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspi…
We introduce a new integral representation for the standard L-function of an irreducible cuspidal automorphic representation of the exceptional group G2, and also for the twist of this L-function by an arbitrary character. Because our construction unfolds to a matrix coefficient rather than a Whittaker function, it applies to non-generic representations as well as generic ones.
Let πbe a cuspidal automorphic representation of GL_4 with central character μ^2. It is known that πhas Shalika period with respect to μif and only if the L-function L^S(s, π, \bigwedge^2\otimesμ^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations π_D of the inner form GL_2(D)(\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for π_D…
Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs through certain cuspidal, automorphic representation of GL(2). Our calculations include the case of square-free level for the p-adic components of tau, and a wide class of archimedean types including Maass forms. As an application we obtain a special value result for L(s,pi x tau).
We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation πof GSp(4,F), where F is a non-archimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in πto be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local L-function of πtwisted by any irreducible, admissible representation of GL(2,F). Together with results in \cite{Fu} and \cite{PS2}, w…
Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods
Institute of Mathematical Sciences, NYU Shanghai（纽约大学上海校区数学科学研究所）；Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité（巴黎左岸儒勒·庞加莱数学研究所，巴黎西岱大学）；Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay（凡尔赛数学实验室，凡尔赛大学，法国国家科学研究中心，巴黎萨克雷大学）；School of Engineering, Mathematics and Physics, University of East Anglia（东英吉利大学工程、数学与物理学院）；Department of Mathematics, Faculty of Science, Kyoto University（京都大学理学部数学科）
Let $π$ be an irreducible cuspidal automorphic generic representation of $\mathrm{Sp}_{2n}(\mathbb{A})$ and let $χ:F^\times\backslash \mathbb{A}^\times\to \mathbb{C}^\times$ be a unitary idele class character. In this note, we present a Rankin-Selberg integral of Hecke type for the twisted standard partial $L$-function $L^S(s,π\timesχ)$ of degree $2n+1$.
Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $σ$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter $(χ,b)$ in the global Arthur parameter $ψ_σ$ associated to $σ$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(χ,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of c…
We consider a new integral representation for $L(s_1, Π\times τ_1) L(s_2, Π\times τ_2),$ where $Π$ is a globally generic cuspidal representation of $GSp_4,$ and $τ_1$ and $τ_2$ are two cuspidal representations of $GL_2$ having the same central character. As and application, we find a new period condition for two such $L$ functions to have a pole simultaneously. This points to an intriguing connection between a Fourier coefficient of a residual representation on $GSO(12)$ and a theta function on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails to unfold completely, but in a way that …
The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $σ$ of a symplectic group $G$ lifts to a cuspidal representation $π= θ_ψ(σ)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $σ$ are linked to periods of automorphic forms in $π$. Consequently, when our results are combined with the Rallis inner product formula in…
A family of global integrals representing a product of tensor product (partial) $L$-functions: $ L^S(s,π\timesτ_1)L^S(s,π\timesτ_2)... L^S(s,π\timesτ_r) $ are established in this paper, where $π$ is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and $τ_1,...,τ_r$ are irreducible unitary cuspidal automorphic representations of $\GL_{a_1},...,\GL_{a_r}$, respectively. When $r=1$ and the classical group is an orthogonal group, this was studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997 and when $π$ is generic and $τ_1,...,τ_r$ are not …
We present an integral representation for the tensor product $L$-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. This is the first in a series of papers, treating…
In this paper we construct a Rankin-Selberg integral which represents the Spin_10 x St L-function attached to the group GSO_10 x PGL_2. We use this integral representation to give some equivalent conditions for a generic cuspidal representation on GSO_10 to be a functorial lift from the group G_2 x PGL_2.
Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $π$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is $GL(n,F)$-distinguished. In this paper we solve the problem of characterising the occurrence of a pole at $X=1$ of $L^+(X,π)$ when $π$ is an $\ell$-modular cuspidal representation of $GL(n,E)$: we show that $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is a relatively banal distinguished representation; namely $π$…
In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one re…
We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $\mathrm{E}_6$, and relate this period to the Ginzburg-Rallis period of cusp forms on $\mathrm{GL}_6$. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of $\mathrm{GL}_6$
We present a Rankin-Selberg integral on the exceptional group $G_2$ which represents the L-function for generic cuspidal representations of $\widetilde{SL}_2\times GL_2$. As an application, we show that certain Fourier-Jacobi type periods on $G_2$ are non-vanishing.
Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level density to bound the probability of vanishing to order at least $r$ at the central point for families of cuspidal newforms of prime level $N \to \infty$, split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the $n$-level density studied. Previous work determined the optimal test functions for the $1$ and $2$-level densities in certain support ranges, leading to marginal improvements in bounds and making it not a productive av…
Two dimensional adelic objects were introduced by I. Fesenko in his study of the Hasse zeta function associated to a regular model $\mathcal E$ of the elliptic curve $E$. The Hasse-Weil $L$-function $L(E,s)$ of $E$ appears in the denominator of the Hasse zeta function of $\mathcal E$. The two dimensional adelic analysis predicts that the integrand $h$ of the boundary term of the two dimensional zeta integral attached to $\mathcal E$ is mean-periodic. The mean-periodicity of $h$ implies the meromorphic continuation and the functional equation of $L(E,s)$. On the other hand, if $E$ is modular, s…
The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of Harder and Hida (independently). What is different is an organizational principle based on the period relations proved by Raghuram and Shahidi for periods attached to regular algebraic cuspidal automorphic representations. The point of view taken in this article is that one need only prove an algebraicity theorem for the most interesting L-value, namely, the c…
We establish the existence of congruences between a fixed endoscopic cuspidal automorphic representation $Π$ of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at $1$ of the adjoint $L$-function of $Π$ normalized by a suitable period.
For automorphic representations in the nontempered cuspidal spectrum of $\mathrm{SO}_5$, we prove the refined Gan-Gross-Prasad conjecture by establishing a precise Bessel period formula, in which the square of the global Bessel period is expressed as an Euler product of regularized local Bessel period integrals of matrix coefficients.
Let $π$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ which satisfies certain reasonable assumptions such as integrality of Hecke polynomials, the existence of mod $\ell$ Galois representations attached to $π$. Under Langlands functoriality of exterior $m$-th power $\wedge^m(π)$, $m=2,...,[\frac n2]$, we will construct a unique Artin representation associated to $π$. As a corollary, we obtain that such a cuspidal representation of $GL_n(\mathbb{A}_\mathbb{Q})$ satisfies the Ramanujan conjecture. We also revisit our previous work on Artin representations associated t…
Betti-Whittaker periods of the contragredient representations for $\textrm{GL}(n)$
Zhejiang University（浙江大学）；Institute for Advanced Study in Mathematics, Zhejiang University（浙江大学数学高等研究院）；School of Mathematical Sciences, Zhejiang University（浙江大学数学学院）；Institute for Advanced Study in Mathematics and New Cornerstone Science Laboratory, Zhejiang University（浙江大学数学高等研究院与新Cornerstone科学实验室）
The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $π$ of $G_2(\mathbb{A})$ that corresponds to a modular form $φ$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $Λ(π,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $φ$ is nonzero, we prove the functional equation $Λ(π,\operatorname{Std},s) = Λ(π,\operatorname{Std},1-s)$.…
We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $π$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $Λ(π,Spin,s)$ of $π$.
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
Let $π$ be the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $τ$ be an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving the unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,π\timesτ)$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $π$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined wit…
We consider distinction of representations in the context of $p$-adic Galois symmetric spaces. We provide new sufficient conditions for distinction of parabolically induced representations in terms of similar conditions on the inducing data and deduce a characterization for distinction of representations parabolically induced from cuspidal. We explicate the results further for classical groups and give several applications, in particular, concerning the preservation of distinction via Langlands functoriality. We relate our results with a conjecture of Dipendra Prasad.
In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $Sp_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$ for $F$ a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of $U_{2n}({\mathbb A}_k)$ with nonzero period in…
We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on $\mathrm{GSp}_6$. We deduce that for a wide class of cuspidal automorphic representations $π,$ the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $π$ is an exceptional theta lift from $\mathrm{G}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $π$ upon which the aforementioned …
We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character, and express it in terms of a degenerate Eisenstein series. In the local fields analogue, we prove the convergence in a half plane of the local integrals, and their meromorphic continuation. In addition, we find that the unramified calculation gives the Godement-Jacquet zeta function. This realizes and generalizes the construction proposed by Ginzburg and Soud…
In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.
Let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet of $π$ is distinguished if and only if it is $ψ$-generic for a non-degenerate character $ψ$ of $N_n(\mathbb{A}_E)$ trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. We then use this result to analyze the non-vanishing of the period integral on different realizations …
We characterize the cuspidal representations of $G_2$ whose standard $\mathcal{L}$-function admits a pole at $s=2$ as the image of Rallis-Schiffmann lift for the commuting pair $\left(\widetilde{SL_2}, G_2\right)$ in $\widetilde{Sp_{14}}$. The image consists of non-tempered representations. The main tool is the recent construction, by the second author, of a family of Rankin-Selberg integrals representing the standard $\mathcal{L}$-function.
We introduce the notion of finite slope families to encode the local properties of the p-adic families of Galois representations appearing in the work of Harris, Lan, Taylor and Thorne on the construction of Galois representations for (non-self dual) regular algebraic cuspidal automorphic representations of GL(n) over CM fields. Our main result is to prove the analytic continuation of semi-stable (and crystalline) periods for such families.
