We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $π$ be a unitary, cuspidal, automorphic representation of $GL_n(\A_K)$. Let $S$ be a set of finite places of $K$, such that the sum $\sum_{v\in S}Nv^{-2/(n^2+1)}$ is convergent. Then $π$ is uniquely determined by the collection of the local components $\{π_v\mid v\not\in S, ~v \~\text{finite}\}$ of $π$. Combining this theorem with base change, it is possible to consider sets $S$ of positive density, having appropriate splitting behavior with respect to solvable extensions of $K$, and where $π$ is determined upto twisting by a character of the Galois group of $L$ over $K$.
We prove an unconditional power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on GL_2 over any number field which is not totally real. Our proof involves the theory of p-adically completed cohomology developed by Calegari and Emerton, and a bound for the growth of coinvariants in certain finitely generated non-commutative Iwasawa modules.
In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).
Recent results by Abert, Bergeron, Biringer et al., Finis, Lapid and Mueller, and Shin and Templier have extended the limit multiplicity property to quite general classes of groups and sequences of level subgroups. Automorphic representations in the limit multiplicity problem are traditionally counted with multiplicity according to the number of fixed vectors of a level subgroup; our goal is to perform a slightly more refined analysis and count only automorphic representations with a given conductor with multiplicity 1.
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.
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).
Let $F$ be a totally real field, and $\mathbb{A}_F$ be the adele ring of $F$. Let us fix $N$ to be a positive integer. Let $π_1=\otimesπ_{1,v}$ and $π_2=\otimesπ_{2,v}$ be distinct cohomological cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{F})$ with levels less than or equal to $N$. Let $\mathcal{N}(π_1,π_2)$ be the minimum of the absolute norm of $v \nmid \infty$ such that $π_{1,v} \not \simeq π_{2,v}$ and that $π_{1,v}$ and $π_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $π_1$ and $π_2$, $$\mathcal{N}(π_1,π_2) \leq C_N.$$ T…
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.
Let $K$ be an algebraic number field, and $π=\otimesπ_{v}$ an irreducible, automorphic, cuspidal representation of $\GL_{m}(\mathbb{A}_{K})$ with analytic conductor $C(π)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $π$ can be decided completely by its local components $π_{v}$ with norm $N(v)<c\cdot C(π)^{2m+\varepsilon}.$
We prove the classification of discrete automorphic representations of GSp$_4$ explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp$_4$ and Sp$_4$ .
Langlands has described the irreducible admissible representations of $T$, when $T$ is the group of points of an algebraic torus over a local field. Also, Langlands described the automorphic representations of $T_{\mathbb A}$ when $T_{\mathbb A}$ is the group of adelic points of an algebraic torus over a global field $F$. We describe irreducible (in the local setting) and automorphic (in the global setting) $ε$-genuine representations for covers of tori, also known as metaplectic tori, which arise from a framework of Brylinski and Deligne. In particular, our results include a description of sp…
Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm{GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal{A}_{\mathrm{St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of $\mathcal{A}_{\mathrm{St}}$ as automorphic representation an…
Let G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove Arthur's multiplicity formula for automorphic representations of G having algebraic regular infinitesimal character at all places in S.
We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
This paper examines fields of rationality in families of cuspidal automorphic representations of unitary groups. Specifically, for a fixed $A$ and a sufficiently large family $\mathcal{F}$, a small proportion of representations $π\in \mathcal{F}$ will satisfy $[\mathbb{Q}(π):\mathbb{Q}] \leq A$. Like earlier work of Shin and Templier, the result depends on a Plancherel equidistribution result for the local components of representations in families. An innovation of our work is an upper bound on the number of discrete series $GL_n(L)$ representations with small field of rationality, counted wit…
We set up a general framework to compute the exact multiplicity with which certain automorphic representations appear in both the cuspidal and Eisenstein cohomology of locally symmetric spaces. We apply this machinery to Eisenstein series on $GSp_4$ and split $G_2$. In the case of $G_2$, we will also obtain new information about the archimedean components of certain CAP representations using Arthur's conjectures.
Many results are known regarding how much local information is required to determine a global object, such as a modular form, or a Galois or automorphic representation. We survey some things that are known and expected, and then explain recent joint work with Dinakar Ramakrishnan about comparing degree 2 Artin and automorphic representations which a priori may not correspond at certain infinite sets of places.
We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GL(n) x GL(n'). Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of $L(s,f x f~), on the residue at s=1. As an application we show that a cuspidal automorphic representation on GL(n) is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.
Let $π=\otimesπ_{v}$ and $π^{\prime}=\otimesπ_{v}^{\prime}$ be two irreducible, automorphic, cuspidal representations of $GL_{m}(\mathbb{A}_{K}) >.$ Using the logarithmic zero-free region of Rankin-Selberg $L$-function, Moreno established the analytic strong multiplicity one theorem if at least one of them is self-contragredient, i.e. $π$ and $π^{\prime}$ will be equal if they have finitely many same local components $π_{v},π_{v}^{\prime},$ for which the norm of places are bounded polynomially by the analytic conductor of these cuspidal representations. Without the assumption of the self-contr…
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 provide a construction of local and automorphic non-tempered Arthur packets of the group SO(3,2) and its inner form SO(4,1) associated with a certain Arthur's parameter and prove a multiplicity formula. We further study the restriction of the representations in these packets to the subgroup SO(3,1). In particular, we discover that the local Gross-Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1) x SO(3,1) and SO(3,2) x SO(3,1) for the representatio…
Publications on automorphic representations of the group U(3) assumed the validity of multiplicity one theorem since I claimed it in 1982. But the argument, published 1988, was based on a misinterpretation of a claim of Gelbart and Piatetski-Shapiro, SLN 1041 (1984), Prop. 2.4(i): ``L^2_{0,1} has multiplicity 1'', as meaning that each irreducible in the space L^2_{0,1} of generic cusp forms has multiplicity one in the space L^2_0 of cusp forms. The statement meant in SLN 1041, ``each irreducible in L^2_{0,1} has multiplicity 1 in L^2_{0,1}'' is too weak to be useful for nongeneric representati…
For distinct unitary cuspidal automorphic representations $π_1$ and $π_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $α\in\Bbb{R}$, let $\mathcal{S}_α$ be the set of primes $v$ of $F$ for which $λ_{π_1}(v)\neq e^{iα} λ_{π_2}(v)$, where $λ_{π_i}(v)$ is the Fourier coefficient of $π_i$ at $v$. In this article, we show that the lower Dirichlet density of $\mathcal{S}_α$ is at least $\frac{1}{16}$. Moreover, if $π_1$ and $π_2$ are not twist-equivalent, we show that the lower Dirichlet densities of $\mathcal{S}_α$ and $ \cap_α\mathcal{S}_α$ are at least $\frac{2}{13}$ and $\frac{1}{11}$, …
We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G_2 in the cohomology of Siegel modular varieties of genus three. Under some additional local hypotheses we also show automorp…
We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on…
Let F be a number field with adele ring A_F, and πan isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional representation σ_v of the Weil group W_{F_v} attached to π_v is multiplicity free. Suppose there is an associated 4-dimensional, Hodge-Tate p-adic representation ρof the absolute Galois group G_F, whose local L-factors agree with those of π(up to a shift) at almost all primes P of F. Then our first result is that the semisimplification of ρdoes not contain any irr…
Let $k$ be a number field and $\mathbb{A}$ be its ring of adeles. Let $U$ be a unipotent group defined over $k$, and $σ$ a $k$-rational involution of $U$ with fixed points $U^+$. As a consequence of the results of C. Moore, the space $L^2(U(k)\backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^+:ϕ\mapsto \int_{U^+(k)\backslash {U}_{\mathbb{A}}^+} ϕ(u)du$ to be the period integral attached to $σ$ on the space of smooth vectors of $L^2(U(k)\backslash U_{\mathbb{A}})$, we prove that if $Π$ is a topologically irreducible subspace of $L^2(U(k)\backsl…
It is known that multiplicity one property holds for SL(2), while the strong multiplicity one property fails. However, in this paper, we show that if we require further that a pair of cuspidal representations $π$ and $π'$ of SL(2) have the same local components at archimedean places and the places above 2, and they are generic with respect to the same additive character, then they also satisfy the strong multiplicity one property. The proof is based on a local converse theorem for SL(2).
We study the problem of determining elements of the Selberg class by information on the coefficents of the Dirichlet series at the squares of primes, or information about the zeroes of the functions.
A fundamental question, first raised by Langlands, is to know whether the Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose standard L-function equals L^*(s, f x g) after removing the ramified and archimedean factors. The first main result of this paper is to answer it in the affirmative, in fact with the base field Q replaced by any number field F. Our proof uses a mixture of converse theorems, base change and descent, and it also appeals to the local regularity properties of Eisenstein…
In this note we investigate the problem of determining elements of the Selberg class from their Dirichlet series coefficients at the primes. We show that this is possible when the degree is one, but in general need an additional weak hypothesis on the Dirichlet series coefficients at squares of primes.
Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of $\operatorname{Bun}_G$. It is independent of the work of Lafforgue on the global Langlands correspondence.
Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic
We give two applications of Arthur's multiplicity formula to Siegel modular forms. The one is a lifting theorem for vector valued Siegel modular forms, which contains Miyawaki's conjectures and Ibukiyama's conjectures. The other is the strong multiplicity one theorem for Siegel modular forms of scalar weights and level one.
Let $X$ denote the non-compact globally Hermitian symmetric space of type $DIII$, namely, $\text{SO}(n,\mathbb{H})/\text{U}(n)$. Let $Λ$ be a uniform torsionless lattice in $\text{SO}(n,\mathbb{H})$. In this note we construct certain complex analytic submanifolds in the locally symmetric space $X_Γ:=Γ\backslash \text{SO}(n,\mathbb{H})/\text{U}(n)$ for certain finite index sub lattices $Γ\subset Λ$ and show that their dual cohomology classes in $H^*(X_Γ;\mathbb{C})$ are not in the image of the Matsushima homomorphism $H^*(X_u; \mathbb{C})\to H^*(X_Γ;\mathbb{C})$, where $X_u=\text{SO}(2n)/\text{…
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 …
In this note we study the symmetric powers of strongly modular icosahedral representations $ρ$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $ρ$, there exists a cuspidal automorphic representation $Π= Π_{\infty} \otimes Π_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (ρ)) = L (s, Π_{f})$. One sees that ${\rm sym}^{5} (ρ)$ is twist equivalent to $ρ' \otimes {\rm sym}^{2} (ρ)$ for another modular icosahedral representation $ρ'$, and our theorem is a special case of a cuspidality criterion formulated and proved in this …
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
Let $L$ be a finite extension of $\mathbb{Q}_p$ and $n\geq 2$. We associate to a crystabelline $n$-dimensional representation of $\mathrm{Gal}(\overline L/L)$ satisfying mild genericity assumptions a finite length locally $\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [BHS17a] we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain "ordinary" $p$-adic Banach space rep…
Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively, multiplicities of lengths of closed geodesics in $\calM_1$ and $\calM_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.
We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifold…
