Automorphic Representations Research Articles

The standard sign conjecture on algebraic cycles: The case of Shimura varieties

Abstract We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur’s conjectures plus a bit more). We also indicate what is known (to us) about these statements.

On the Hecke fields of Galois representations

It is known that, in many cases, the field (known as the Hecke field) generated by all Hecke eigenvalues of an elliptic modular newform f is in fact generated by the pth Hecke eigenvalue ap of f for a single prime number p. In this paper, we study the density of such primes in a more general situation in terms of Galois representations. This is applied to the study of the fields of rationality of certain automorphic representations.

On the Langlands correspondence for symplectic motives

We present a refinement of the global Langlands correspondence for symplectic motives. Using the local theory of generic representations of odd orthogonal groups, we define a new vector in the associated automorphic representation, which is the tensor product of test vectors for the Whittaker functionals.

On Properties of Certain Special Zeros of Functions in the Selberg Class

In this paper, assuming generalized Riemann hypothesis, we give an upper bound for the multiplicity of eventual zero at central point 1 / 2 and location of the first zero with positive imaginary part of function in a certain subclass of the extended Selberg class. We apply our results to automorphic L-functions attached to irreducible unitary automorphic representations of $$GL_N(\mathbb {Q})$$ .

Archimedean zeta integrals on U(n,1)

For a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions. In this paper we explicitly calculate archimedean zeta integrals of this type for U(n,1), assuming that the corresponding archimedean component of the automorphic representation is a holomorphic discrete series.

Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree n of compact congruence p-dimensional hyperbolic manifolds “of simple type” as long as n is strictly smaller than 1 2 [ p 2 ]. We also prove that for connected Shimura varieties associated to O(p, 2) the Hodge conjecture is true for classes of degree < 1 2 [ p+1 2 ]. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by [3]. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups GL(N)o〈θ〉 and SO(2n) o 〈θ′〉 (where θ and θ′ are the outer automorphisms), see [3, Hypothesis 3.2]. Unfortunately, at present the stabilization of the trace formula has been proved only for the case of connected groups. The extension needed is part of work in progress by the Paris-Marseille team of automorphic form researchers. For more detail, see the second paragraph of subsection 1.18 below.

Solvable base change and Rankin-Selberg convolutions

Let π and π′ be unitary automorphic cuspidal representations of GLn(AE) and GLm(AF), and let E and F be solvable Galois extensions of Q of degrees l and l′, respectively. Using the fact that the automorphic induction and base change maps exist for E and F, and assuming an invariance condition under the actions of the Galois groups, we attach to the pair (π, π′) a Rankin-Selberg L-function L(s, π ×E,F xxxxxx) for which we prove a prime number theorem. This gives a method for comparing two representations that could be defined over completely different extensions, and the main results give a measure of how many cuspidal components the two representations π and π′ have in common when automorphically induced down to the rational numbers. The proof uses the structure of the Galois group of the composite extension EF and the character groups attached to the fields via class field theory. The second main theorem also gives an indication of when the base change of π up to the composite extension EF remains cuspidal.

On special Bessel periods and the Gross–Prasad conjecture for $$\mathrm {SO}(2n+1) \times \mathrm {SO}(2)$$

In this paper we investigate one instance of the global Gross–Prasad conjecture. Our main result proves that if an irreducible cuspidal automorphic representation \(\pi \) of an odd dimensional special orthogonal group, whose local component \(\pi _{w}\) at some finite place w is generic, admits the special Bessel model corresponding to a quadratic extension E over a base field F, then the central L-value \(L(1/2,\pi )L(1/2,\pi \times \chi _E)\) does not vanish. Here \(\chi _E\) denotes the quadratic character of \(\mathbb A_F^\times \) corresponding to E. As an application, we obtain the equivalence between the non-vanishing of the special Bessel period and that of the corresponding central L-value when \(\pi \) is associated to a full modular holomorphic Siegel cusp form of degree two, which is a Hecke eigenform, and E is an imaginary quadratic extension of \(\mathbb Q\).

Global Jacquet-Langlands Correspondence for Division Algebras in Characteristic $p$

We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from the set of automorphic square integrable representations of the multiplicative group of $D$ to the set of automorphic square integrable representations of GL_n(F), compatible at all places with the local Jacquet-Langlands correspondence for unitary representations. We characterize the image of the map. As a consequence we get multiplicity one and strong multiplicity one theorems for the multiplicative group of D.

A Family of New-Way Integrals for the Standard ℒ-function of Cuspidal Representations of the Exceptional Group of TypeG2

Let |$\mathcal{L}^{S}\!\left({s,\pi,\chi,\operatorname{\mathfrak{st}}}\right)\!$| be a standard twisted partial |$\mathcal{L}$|-function of degree |$7$| of the cuspidal automorphic representation |$\pi$| of the exceptional group of type |$G_2$|⁠. In this paper, we consider a family of Rankin–Selberg integrals and prove that it represents this |$\mathcal{L}$|-function. As an application, we prove that the representations attaining certain prescribed poles are exactly the representations attained by |$\theta$|-lift from a group of\break finite type.

On the image of complex conjugation in certain Galois representations

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

On the Genericity of Eisenstein Series and Their Residues for Covers of $GL_m$

Let $\tau_1^{(r)}$, $\tau_2^{(r)}$ be two genuine cuspidal automorphic representations on $r$-fold covers of the adelic points of the general linear groups $GL_{n_1}$, $GL_{n_2}$, resp., and let $E(g,s)$ be the associated Eisenstein series on an $r$-fold cover of $GL_{n_1+n_2}$. Then the value or residue at any point $s=s_0$ of $E(g,s)$ is an automorphic form, and generates an automorphic representation. In this note we show that if $n_1\neq n_2$ these automorphic representations (when not identically zero) are generic, while if $n_1=n_2:=n$ they are generic except for residues at $s=\frac{n\pm1}{2n}$.

Best possible rates of distribution of dense lattice orbits in homogeneous spaces

Abstract This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group G, acting on suitable homogeneous spaces G/H. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G/Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on L 2 L^{2} (Γ ∖ \setminus G). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

On tensor third $L$-functions of automorphic representations of $GL_n(\mathbb {A}_F)$

Langlands’ beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,\pi , \otimes ^3),$ where $\pi$ is a cuspidal automorphic representation of $\mathrm {GL}_n(\mathbb {A}_F)$ and $F$ is a global field.

A general framework of automorphic inflation

Automorphic inflation is an application of the framework of automorphic scalar field theory, based on the theory of automorphic forms and representations. In this paper the general framework of automorphic and modular inflation is described in some detail, with emphasis on the resulting stratification of the space of scalar field theories in terms of the group theoretic data associated to the shift symmetry, as well as the automorphic data that specifies the potential. The class of theories based on Eisenstein series provides a natural generalization of the model of $j$-inflation considered previously.

A converse theorem for GL(n)

We complete the work of Cogdell and Piatetski-Shapiro [3] to prove, for n≥3, a converse theorem for automorphic representations of GLn over a number field, with analytic data from twists by unramified representations of GLn−1.

Small automorphic representations and degenerate Whittaker vectors

We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand–Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also use our formalism to calculate various local (real and p-adic) spherical vectors of minimal representations of the exceptional groups E6, E7, E8 using global (adelic) degenerate Whittaker vectors, correctly reproducing existing results for such spherical vectors obtained by very different methods.

Distribution of rational points of bounded height on equivariant compactifications of PGL 2 I

We study the distribution of rational points of bounded height on a one-sided equivariant compactification of PGL2 using automorphic representation theory of PGL2.

Asymptotics for cuspidal representations by functoriality from GL(2)

Let π be a unitary automorphic cuspidal representation of GL2(QA) with Fourier coefficients λπ(n). Asymptotic expansions of certain sums of λπ(n) are proved using known functorial liftings from GL2, including symmetric powers, isobaric products, exterior squares, and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of λπ(n) on integers, squares, cubes and fourth powers.

On the geometric construction of cohomology classes for cocompact discrete subgroups of SLn(ℝ) and SLn(ℂ)

We construct nontrivial cohomology classes for certain cocompact discrete subgroups of SLn.R/ and SLn.C/ using a geometric method. The discrete subgroups are of arithmetic nature, i.e., they arise from arithmetic subgroups of suitably chosen algebraic groups. In certain cases, we show the nonvanishing of automorphic representations as a consequence.