Cuspidal Automorphic Representations Research Articles

Period Relations Between the Betti–Whittaker Periods for GLn Under Duality

Abstract In this paper, under some regularity conditions, we prove a period relation between the Betti–Whittaker periods associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{n}({{\mathbb{A}}})$ and its contragredient. As a consequence, we prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{2n}({{\mathbb{A}}})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for $\textrm{GSpin}_{2n}^{*} \times \operatorname{GL}_{n^{\prime}}$ by the result of Harder and Raghuram.

Zeros of Rankin–Selberg L-functions in families

Let $\mathfrak {F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm {GL}_n$ with unitary central character over a number field $F$ . We prove the first unconditional zero density estimate for the set $\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg $L$ -functions, where $\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at $s=\frac {1}{2}$ for almost all $L(s,\pi \times \pi ')\in \mathcal {S}$ ; (ii) a strong on-average form of effective multiplicity one for almost all $\pi \in \mathfrak {F}_n$ ; and (iii) a positive level of distribution for $L(s,\pi \times \widetilde {\pi })$ , in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$ .

Value Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic L-functions

Abstract Let $d\in\mathbb{N}$ and π be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L^{\prime}}{L}(1+it,\pi\otimes\chi_D)$ as D varies over fundamental discriminants. Here, t is a fixed real number and χD is the real character associated with D. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L^{\prime}}{L}(1,\pi\otimes\chi_D)\right|$ when π is self-dual.

Local-global compatibility for regular algebraic cuspidal automorphic representations when

Abstract We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$ . We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$ , the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$ .

GALOIS REPRESENTATIONS FOR EVEN GENERAL SPECIAL ORTHOGONAL GROUPS

Abstract We prove the existence of $\mathrm {GSpin}_{2n}$ -valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$ , arising from forms of ${\mathrm {GSO}}_{2n}$ . As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$ -valued Galois representations by removing the outer automorphism ambiguity.

Anticyclotomic Euler systems for unitary groups

AbstractLet be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature .

The second moment of Dirichlet twists of a text {GL}_{4} automorphic L-function

In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic L-function L(s, pi ) on the critical line averaged over characters and conductors, where pi denotes an irreducible tempered cuspidal automorphic representation of {textrm{GL}}_{4}({mathbb {A}}_{mathbb {Q}}) 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.

On the Whittaker range of the generalized metaplectic theta lift

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. In this work we study the existence of generic lifts in the resulting theta tower. In the classical case, there are two orthogonal groups that may support a generic lift of an irreducible cuspidal automorphic representation of a symplectic group. We show that in general the Whittaker range consists of r+1 groups for the lift from the r-fold cover of a symplectic group. We also give a period criterion for the genericity of the lift at each step of the tower.

Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

AbstractLet be a smooth, separated, geometrically connected scheme defined over a number field and a system of semisimple ‐adic representations of the étale fundamental group of such that for each closed point of , the specialization is a compatible system of Galois representations under mild local conditions. For almost all , we prove that any type A irreducible subrepresentation of is residually irreducible. When is totally real or CM, , and is the compatible system of Galois representations of attached to a regular algebraic, polarized, cuspidal automorphic representation of , for almost all , we prove that is (i) irreducible and (ii) residually irreducible if in addition .

Automorphic descent for symplectic groups: The branching problems and L -functions

We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for ${\rm Mp}_2(\Bbb{A})$, we give a new approach to prove the result that for an automorphic cuspidal representation of ${\rm GL}_2(\Bbb{A})$ of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central $L$-values.

Dimension formulas for Siegel modular forms of level 4

AbstractWe prove several dimension formulas for spaces of scalar‐valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of whose local component at admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at , we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.

Endoscopic congruences modulo adjoint L-values for GSp(4)

We establish the existence of congruences between a fixed endoscopic, globally generic, cuspidal automorphic representation Π of GSp(4) of square-free conductor and stable cuspidal automorphic representations of the same weight modulo certain prime factors of the value at 1 of the adjoint L-function of Π, normalized by a suitable period.

Potential automorphy over CM fields

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbb{A}_F)$.

The Weyl bound for triple product L-functions

Let π1, π2, π3 be three cuspidal automorphic representations for the group SL(2,Z), where π1 and π2 are fixed and π3 has large analytic conductor. We prove a subconvex bound for L(1∕2,π1⊗π2⊗π3) of Weyl-type quality. Allowing π3 to be an Eisenstein series, we also obtain a Weyl-type subconvex bound for L(1∕2+it,π1⊗π2).

Distinction inside L-packets of SL(n)

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $\pi$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(\sigma,d)$ be the unique pair associated to $\pi$, where $\sigma$ is a cuspidal representation of $\mathrm{GL}_r(\mathbb{A}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $\pi$ is distinguished with respect to $\mathrm{SL}_n(\mathbb{A}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $\psi$ of type $r^d:=(r,\dots,r)$ of $N_n(\mathbb{A}_E)$ which is trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. As a first application, under the assumptions that $E/F$ splits at infinity and $r$ is odd, we establish a local-global principle for $\mathrm{SL}_n(\mathbb{A}_F)$-distinction inside the L-packet of $\pi$. As a second application we construct examples of distinguished cuspidal automorphic representations $\pi$ of $\mathrm{SL}_n(\mathbb{A}_E)$ such that the period integral vanishes on some canonical copy of $\pi$, and of everywhere locally distinguished representations of $\mathrm{SL}_n(\mathbb{A}_E)$ such that their L-packets do not contain any distinguished representation.

Automorphic tensor products and cuspidal cohomology of the GL4

In this paper, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of [Formula: see text] to cuspidal cohomology of the [Formula: see text] which are obtained from automorphic tensor product of two automorphic representations of [Formula: see text] of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of [Formula: see text] and the automorphic tensor product of two representations of [Formula: see text] cannot be equal (up to a twist by a character of [Formula: see text]) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.

On Deligne's conjecture for symmetric fourth L-functions of Hilbert modular forms

We prove an automorphic analogue of Deligne's conjecture for symmetric fourth L-functions of Hilbert modular forms. We extend the result of Morimoto [41] based on generalization and refinement of the results of Grobner and Lin [18] to cohomological irreducible essentially conjugate self-dual cuspidal automorphic representations of GL2 and GL3 over CM-fields.

A coarse geometric expansion of a variant of Arthur’s truncated traces and some applications

Let F be a global function field with constant field $\mathbb{F}_q$. Let G be a reductive group over $\mathbb{F}_q$. We establish a variant of Arthur's truncated kernel for G and for its Lie algebra which generalizes Arthur's original construction. We establish a coarse geometric expansion for our variant truncation. As applications, we consider some existence and uniqueness problems of some cuspidal automorphic representations for the functions field of the projective line $\mathbb{P}^1_{\mathbb{F}_q}$ with two points of ramifications.

On automorphic descent from $GL_7$ to $G_2$

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations $\pi$ of $GL\_7(\mathbb{A})$ with $L^S(s, \pi, \wedge^3)$ having a pole at $s=1$ to the split exceptional group $G\_2(\mathbb{A})$, using Fourier coefficients associated to two nilpotent orbits of $E\_7$. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and $\pi$ is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube $L$-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on $G\_2(\mathbb{A})$ but also every degenerate Whittaker–Fourier integral. Thus it is generic, but not cuspidal.

The completed standard L-function of modular forms on $$G_2$$

The goal of this paper is to provide a complete and refined study of the standard L-functions $$L(\pi ,{\text {Std}},s)$$ for certain non-generic cuspidal automorphic representations $$\pi $$ of $$G_2({\mathbb {A}})$$ . For a cuspidal automorphic representation $$\pi $$ of $$G_2({\mathbb {A}})$$ that corresponds to a modular form $$\varphi $$ of level one and of even weight on $$G_2$$ , we explicitly define the completed standard L-function, $$\Lambda (\pi ,{\text {Std}},s)$$ . Assuming that a certain Fourier coefficient of $$\varphi $$ is nonzero, we prove the functional equation $$\Lambda (\pi ,{\text {Std}},s) = \Lambda (\pi ,{\text {Std}},1-s)$$ . Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.