Automorphic Representations Research Articles

Estimates for coefficients of certain L-functions

Let $$\pi _{\varphi }$$ (or $$\pi _{\psi }$$ ) be an automorphic cuspidal representation of $$\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})$$ associated to a primitive Maass cusp form $$\varphi $$ (or $$\psi $$ ), and $$\mathrm{sym}^j \pi _{\varphi }$$ be the jth symmetric power lift of $$\pi _{\varphi }$$ . Let $$a_{\mathrm{sym}^j \pi _{\varphi }}(n)$$ denote the nth Dirichlet series coefficient of the principal L-function associated to $$\mathrm{sym}^j \pi _{\varphi }$$ . In this paper, we study first moments of Dirichlet series coefficients of automorphic representations $$\mathrm{sym}^3 \pi _{\varphi }$$ of $$\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})$$ , and $$\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }$$ of $$\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})$$ . For $$3 \le j \le 8$$ , estimates for $$|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|$$ on average over a short interval have also been established.

On Fourier coefficients of certain residual representations of symplectic groups

In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays important roles. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, corresponding to certain interesting families of global Arthur parameters. On one hand, the results partially confirm a conjecture proposed by the first named author in [J14] on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. On the other hand, the results of this paper can be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.

Analytic monoids and factorization problems

The main goal of this paper is to initiate study of analytic monoids as a general framework for quantitative theory of factorization. So far the latter subject was developed either in concrete settings, for instance in orders of number fields, or abstractly, in an axiomatic way. Some of the abstract approaches are too general to address delicate problems concerning oscillatory nature of the related counting functions, or are too restrictive in the sense that they suffer from the lack of examples except classical ones i.e. the Hilbert monoids of algebraic integers and their products. The notion of an analytic monoid is enough flexible to allow constructions of many other examples, and also ensures sufficiently rich analytic structure. In particular, we construct examples of such monoids with the associated L-functions being products of classical Dirichlet L-functions and L-functions of twisted irreducible unitary cuspidal automorphic representations of GL_d({mathbb {A}}_{mathbb {Q}}) satisfying the Ramanujan conjecture and having real coefficients. Finally, to illustrate how a typical problem from the quantitative theory of factorization can be studied in the framework of analytic monoids, we formulate several results concerning oscillations of the remainder term in the asymptotic formula for the number of irreducible elements with norms less or equal x, as x tends to infinity.

Metaplectic Tensor Products for Automorphic Representation of (r)

AbstractLet M = GLr1 ✗ … × GLrk ⊆ GLr be a Levi subgroup of GLr, where r = r1 + … +rk, and its metaplectic preimage in the n-fold metaplectic cover r1 of GLr. For automorphic representations π1, …, πk of r1 (), … ,rk (), we construct (under a certain technical assumption that is always satisfied when n = 2) an automorphic representation π of () that can be considered as the “tensor product” of the representations π1, … , πk. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, πv is equivalent to the local metaplectic tensor product of π1,v, … , πk,v defined by Mezo. Then we show that if all of the πi are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element and show the compatibility with parabolic inductions.

Zero correlation with lower-order terms for automorphic L-functions

Let [Formula: see text] be a self-contragredient automorphic cuspidal representation of [Formula: see text] for [Formula: see text]. Using a refined version of the Selberg orthogonality, we recompute the [Formula: see text]-level correlation of high non-trivial zeros of the product [Formula: see text]. In the process, we are able to extract certain low-order terms which suggest the asymptotics of these statistics are not necessarily universal, but depend upon the conductors of the representations and hence the ramification properties of the local components coming from each [Formula: see text]. The computation of these lower-order terms is unconditional as long as all [Formula: see text].

Representations of toral automorphisms

This survey gives an account of an algebraic construction of symbolic covers and representations of ergodic automorphisms of compact abelian groups, initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of finite-dimensional tori. The key ingredient in this approach, which was subsequently extended to arbitrary expansive automorphisms of compact abelian groups, is the use of homoclinic points of the automorphism.Although existence and abundance of homoclinic points is intimately connected to expansiveness of the automorphism, it is nevertheless possible to extend certain aspects of this construction to nonexpansive irreducible automorphisms of compact abelian groups (like irreducible toral automorphisms whose dominant eigenvalue is a Salem number). The later sections of this survey discuss the phenomena and problems arising in this extension.

Some conjectures on endoscopic representations in odd orthogonal groups

In this paper, we introduce two conjectures on characterizations of endoscopy structures of irreducible generic cuspidal automorphic representations of odd special orthogonal groups in terms of nonvanishing of certain period of automorphic forms. We discuss a relation between the two conjectures and prove that a special case of Conjecture 1 (and hence Conjecture 2) is true.

Epipelagic representations and rigid local systems

We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ng\^o and the author. We study the monodromy of these local systems and compute all examples when $G$ is a classical group.

Quadratic base change and the analytic continuation of the Asai L-function: A new trace formula approach

Using Langlands's beyond endoscopy idea, we study the Asai $L$-function associated to a real quadratic field $\Bbb{K}/\Bbb{Q}$. We prove that the Asai $L$-function associated to a cuspidal automorphic representation over $\Bbb{K}$ has analytic continuation to the complex plane with at most a simple pole at $s=1$. We then show if the $L$-function has a pole then the representation is a base change from $\Bbb{Q}$. While this result is known using integral representations from the work of Asai and Flicker, the approach here uses novel analytic number techniques and gives a deeper understanding of the geometric side of the relative trace formula. Moreover, the approach in this paper will make it easier to grasp more complicated functoriality via beyond endoscopy. We also describe a connection of Langlands's technique to the distinguishing of representations via periods.

The Hodge conjecture and arithmetic quotients of complex balls

Let S be a closed Shimura variety uniformized by the complex n-ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in \({H^{2k} (S, \mathbb{Q})}\), \({k=0,\dots, n}\), is algebraic. We show that this holds for all degrees k away from the neighborhood \({\bigl]\tfrac13n,\tfrac23n\bigr[}\) of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups \({GL (N) \rtimes \langle \theta \rangle}\) associated with base change.

Period relations for automorphic forms on unitary groups and critical values of $L$-functions

In this paper we explore some properties of periods attached to automorphic representations of unitary groups over CM fields and the critical values of their L -functions. We prove a formula expressing the critical values in the range of absolute convergence in terms of Petersson norms of holomorphic automorphic forms. On the other hand, we express the Deligne period of a related motive as a product of quadratic periods and compare the two expressions by means of Deligne's conjecture.

On the ramification of modular parametrizations at the cusps

We investigate the ramification of modular parametrizations of elliptic curves over Q at the cusps. We prove that if the modular form associated to the elliptic curve has minimal level among its twists by Dirichlet characters, then the modular parametrization is unramified at the cusps. The proof uses Bushnell's formula for the Godement-Jacquet local constant of a cuspidal automorphic representation of GL(2). We also report on numerical computations indicating that in general, the ramification index at a cusp seems to be a divisor of 24.

Covers of tori over local and global fields

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 $T_{\Bbb{A}}$ when $T_{\Bbb{A}}$ 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) $\epsilon$-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 spherical Hecke algebras in the local unramified setting, and a global multiplicity estimate for automorphic representations of covers of split tori. For automorphic representations of covers of split tori, we prove a multiplicity-one theorem.

$p$-adic L-functions of automorphic forms and exceptional zeros

We construct p -adic L-functions for automorphic representations of \mathrm{GL} _2 of a number field F , and show that the corresponding p -adic L-function of a modular elliptic curve E over F has an extra zero at the central point for each prime above p at which E has split multiplicative reduction, a part of the exceptional zero conjecture.

On special values of certain L-functions, II

We prove an algebraicity result concerning special values at critical points, in the sense of Deligne, of tensor product $L$-functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite, and, cuspidal representations of ${\rm GL}(2)$ corresponding to primitive cusp forms, over totally real number fields. We also prove the reciprocity law, i.e., the equivariance under the action of ${\rm Gal}({\overline{\Bbb Q}}/{\Bbb Q})$, for the special values. In the appendix, the second author calculates the Deligne periods for such $L$-functions, assuming the existence of corresponding motives and the automorphic transfer to a quasi-split form of the special orthogonal group. Our result conforms with the celebrated conjecture of Deligne on special values of motivic $L$-functions.

Eisenstein congruences for split reductive groups

We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases and use the Bloch–Kato conjecture to further motivate a belief in the congruences.

A Specialisation of the Bump–Friedberg L-function

AbstractWe study the restriction of Bump–Friedberg integrals to affine lines {(s + α, 2s), s ∊ ℂ}. It has simple theory, very close to that of the Asai L-function. It is an integral representation of the product L(s + α, π)L(2s, Λ2, π), which we denote by Llin(s, π, α) for this abstract, when π is a cuspidal automorphic representation of GL(k, 𝔸) for 𝔸 the adeles of a number field. When k is even, we show that the partial L-function Llin,S(s, π, α) has a pole at 1/2 if and only if π admits a (twisted) global period. This gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that π has a twisted global period if and only if L(α + 1/2, π) ≠ 0 and L(1, Λ2 , π) = ∞. When k is odd, the partial L-function is holmorphic in a neighbourhood of Re(s) ≥ 1/2 when Re(α) is ≥ 0.

THE DOUBLE COVER OF ODD GENERAL SPIN GROUPS, SMALL REPRESENTATIONS, AND APPLICATIONS

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of$\mathit{SO}_{2n+1}$of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

A nonabelian trace formula

Let E/F be an everywhere unramified extension of number fields with Gal(E/F) simple and nonabelian. In a recent paper, the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of GL2 along such an extension. Motivated by this, we prove a trace formula whose spectral side is a weighted sum over cuspidal automorphic representations of GL 2 ( A E ) $\text {GL}_{2}(\mathbb {A}_{E})$ that are isomorphic to their Gal(E/F)-conjugates. The basic method, which is of interest in itself, is to use functions in a space isolated by Finis, Lapid, and Muller to build more variables into the trace formula. 2010 Mathematics subject classification: Primary 11F70, Secondary 11F66

Exposing relative endoscopy in unitary symmetric spaces

We introduce a new class of symmetric space orbital integrals important for applications in certain relative trace formula appearing in the theory of automorphic representations. We verify a fundamental lemma for U2×U2↪U4 via an explicit calculation, giving the first known example of endoscopy for symmetric spaces and showing strong evidence that there is a general theory of endoscopy lurking in this situation. AMS subject classification Primary 20G05