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.
Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke eigenvalues that are less than an explicit negative constant. This provides an answer to a question of Serre. We also consider analogous problems for cuspidal automorphic representations that are not self-dual.
In this article, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ to cuspidal cohomology of the ${\rm GL}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\rm GL}_2$ and the automorphic tensor product of two representations of ${\rm GL}_2$ can not be equal (up to a twist by a character of ${\rm G…
In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily $K_\infty$-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a ''smooth version'' of the space of automorphic forms, whose internal structure was the topic of a famous paper of Franke. We prove that the important decomposition along the parabolic support, and the even finer - and structurally more important - decomposition along the cuspidal su…
We establish "higher depth" analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic L-functions of GL_d over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.
We find the number $s_k(p,Ω)$ of cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $Ω$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,Ω)$ generalizes to the vector-valued case and a finite number of ramified places.
Let $F$ be a global field. Let $G$ and $H$ be two connected reductive group defined over $F$ endowed with an $F$-morphism $f: H\rightarrow G$ such that the induced morphism $H_{der}\rightarrow G_{der}$ on the derived groups is a central isogeny. Our main results yield in particular the following theorem: Given any irreducible cuspidal representation $π$ of $G(\mathbb A_F)$ its restriction to $H(\mathbb A_F)$ contains a cuspidal representation $σ$ of $H(\mathbb A_F)$. Conversely, assuming moreover that $f$ is an injection, any irreducible cuspidal representation $σ$ of $H(\mathbb A_F)$ appears …
Let $M=GL_{r_1}\times\cdots\times GL_{r_k}\subseteq GL_r$ be a Levi subgroup of $GL_r$, where $r=r_1+\cdots+r_k$, and $\tilde{M}$ its metaplectic preimage in the metaplectic cover $\tilde{GL}_r$ of $GL_r$. For automorphic representations $π_1,\dots,π_k$ of $\tilde{GL}_{r_1}(\A),\dots,\tilde{GL}_{r_k}(\A)$, we construct an automorphic representation $π$ of $\tilde{M}(\A)$ which can be considered as the "tensor product" of the representations $π_1,\dots,π_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $π_v$ is equ…
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…
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 extend some results of Gun, Murty, and Rath on elliptic modular forms. We take ANY Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in the natural parameter have algebraic coefficients. We show that the zeros of such forms are either transcendental or are "CM." By "CM," we mean they correspond to abelian varieties with complex multiplication. This result is the first of its kind in the case of non-arithmetic groups.
Let Π be a cuspidal automorphic representation for GL(4) over a number field F. We obtain unconditional lower bounds on the number of places at which the Satake parameters are not "too large". In the case of self-dual Π with non-trivial central character, our results imply that the set of places at which Π is tempered has an explicit positive lower Dirichlet density. Our methods extend those of Ramakrishnan by careful analysis of the hypothetical possibilities for the structure of the Langlands conjugacy classes, as well as their behaviour under functorial lifts. We then discuss the analogous …
Given a cuspidal automorphic representation $π$ for GL(3) over a number field and a positive integer $k$, assume that the symmetric $m$th power lifts of $π$ are isobaric automorphic for $m \leq k$, cuspidal for $m \leq k-1$, and that certain associated Rankin-Selberg products are isobaric automorphic. Then the number of cuspidal isobaric summands in the $k$th symmetric power lift is bounded above by 3 when $k \geq 7$, and bounded above by 2 when $k \geq 19$ with $k \equiv 1 \pmod 3$. We then investigate the analogous problem for GL(4).
We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(π)$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $π$ of $GL_n(\mathbb{A}_F)$. We show that the restriction of $r_p(π)$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec(π_v)$, the image of $π_v$ under the local Langlands…
We convolve a theta function on an $n$-fold cover of $GL_3$ with an automorphic form on an $n'$-fold cover of $GL_2$ for suitable $n,n'$. To do so, we induce the theta function to the $n$-fold cover of $GL_4$ and use a Shalika integral. We show that in particular when $n=n'=3$ this construction gives a new Eulerian integral for an automorphic form on the 3-fold cover of $GL_2$ (the first such integral was given by Bump and Hoffstein), and when $n=4$, $n'=2$, it gives a Dirichlet series with analytic continuation and functional equation that involves both the Fourier coefficients of an automorp…
Let G be a reductive algebraic group over a number field k. It is shown how Emerton's methods may be applied to the problem of p-adically interpolating the metaplectic forms on G, i.e. the automorphic forms on metaplectic covers of G, as long as the metaplectic covers involved split at the infinite places of k.
In this paper, we consider automorphic forms on $\mathrm{Sp}_4(\mathbb{A}_\mathbb{Q})$ which generate large discrete series representations of $\mathrm{Sp}_4(\mathbb{R})$ as $(\mathfrak{sp}_4(\mathbb{R}),K_\infty)$-modules. We determine the cuspidal components and the structure of the space of such automorphic forms.
The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issu…
There are few constructions of square-integrable automorphic functions on metaplectic groups. Such functions may be obtained by the residues of certain Eisenstein series on covers of groups, "theta functions," but the Fourier coefficients of these residues are not well-understood, even for low degree covers of $GL_2$. Patterson and Chinta-Friedberg-Hoffstein proposed conjectured relations for the Fourier coefficients of the $GL_2$ quartic and sextic theta functions (resp.), each obtained from a conjectured equality of non-Eulerian Dirichlet series. In this article we propose a new framework fo…
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…
In this paper we construct smooth cuspidal automorphic forms related to integrable discrete series of a connected semisimple Lie group with finite center for classical and adelic situation as an application of the theory of Schwartz spaces for automorphic forms developed by Casselman. In the classical situation, smooth cuspidal automorphic forms are constructed via an explicit continuous map from the Frech\' et space of smooth vectors of a Banach realization inside $L^1(G)$ of an integrable discrete series into the space of smooth vectors of a strong topological dual of an appropriate Schwartz…
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
In previous papers we formulated an analogue of the Ichino--Ikeda conjectures for Whittaker--Fourier coefficients of automorphic forms on classical group and the metaplectic group. In the latter case we reduced the conjecture to a local identity. In this paper we will prove the local identity in the $p$-adic case, and hence the global conjecture under simplifying conditions at the archimedean places.
We prove that the complete $L$-function associated to any cuspidal automorphic representation of $GL_2(\mathbb{A}_{\mathbb Q})$ has infinitely many simple zeros.
For a cuspidal automorphic representation Πof GL(4,A), H. Kim proved that the exterior square transfer \wedge^2Πis an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations Πfor which \wedge^2Πis cuspidal.
In this paper we fully describe the cuspidal and the Eisenstein cohomology of the group $G=GL_2$ over a definite quaternion algebra $D/\Q$. Functoriality is used to show the existence of residual and cuspidal automorphic forms, having cohomology in degree 1. The latter ones turn out to be CAP-representations, though $G$ satisfies Strong Multiplicity One. A non-vanishing result on intertwining operators of induced representations will serve as a starting point for further investigations concerning rationality of critical $L$-values.
The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest to vary the orthogonal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. …
For $m\ge 2$, let $π$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_π(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $π$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_π(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $π$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $π$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Se…
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.
We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $π$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group ${\bf G}_n$ and $τ$ a cuspidal (unitary) automorphic representation of a general linear group, then $L(s,π\times τ)$ is holomorphic for $\Re(s) > 1$ and has at most a simple pole at $s=1$. We also prove the holomorphy and non-vanishing of automorphic exterior square, symmetric s…
In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are clo…
Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets, and furthermore state and prove the decay for smooth but not necessarily K-finite cusp forms. We also state a general theorem about the convergence of Rankin-Selberg integrals involving unipotent periods, closing a gap in the literature on L-functions. These properties serve as the analytic basis of a new …
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…
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…
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…
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' fam…
We construct Eigenvarieties for PEL Shimura varieties which interpolate cuspidal, finite slope automorphic forms for PEL Shimura varieties appearing as global sections of (coherent) automorphic sheaves, under the hypothesis that the primes are unramified in the Shimura datum, but allowing the ordinary locus to be empty. We then use this construction to extend previous results on the Bloch-Kato conjecture for Hecke characters of a quadratic imaginary field.
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $ρ$. This generalizes previous work of the author and James Newton.
We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological automorphic forms and a conditional power saving for their multiplicities when the associated locally symmetric space is a hyperbolic 3-manifold.
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.
Let $F$ be a number field, let $\mathbb{A}_F$ be its ring of adeles, and let $g_1,g_2,h_1,h_2 \in \mathrm{GL}_2(\mathbb{A}_F)$. Previously the author provided an absolutely convergent geometric expression for the four variable kernel function $$ \sum_π K_π(g_1,g_2)K_{π^{\vee}}(h_1,h_2)L(s,(π\times π^{\vee})^S), $$ where the sum is over isomorphism classes of cuspidal automorphic representations $π$ of $\mathrm{GL}_2(\mathbb{A}_F)$. Here $K_π$ is the typical kernel function representing the action of a test function on the space of the cuspidal automorphic representation $π$. In this paper we s…
In this paper we generalize the work of Harris-Soudry-Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on GL_2 over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semisimplification.
Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this note, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin-Selberg products imply lower bounds on the size of S. We also obtain further results for GL(3).
In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.
Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$ and $X$ a smooth projective geometrically connected curve over ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we sho…
We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.
In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for general semisimple algebraic group $G$ defined over a number field $k$ such that its Archimedean group $G_\infty$ is not compact. When $G$ is quasi--split over $k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize a result of Vign\' eras, Henniart, and Shahidi. We also discuss the existence of cuspidal automorphic forms with non--zero Fourier coefficients for congruence of subgroups of $G_\infty$.
Let $F$ be a CM field with totally real subfield $F^+$ and let $π$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $π$ a Galois representation $R_π:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_π(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (esse…
We study the behaviour of automorphic L-Invariants associated to cuspidal representations of GL(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic L-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic L-invariants. This together with Spiess' results on p-adic L-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves - at …
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.
