A key ingredient in the Taylor–Wiles proof of Fermat’s last theorem is the classical Ihara lemma, which is used to raise the modularity property between some congruent Galois representations. In their work on Sato and Tate, Clozel, Harris and Taylor proposed a generalisation of the Ihara lemma in higher dimension for some similitude groups. The main aim of this paper is to prove some new instances of this generalised Ihara lemma by considering some particular non-pseudo-Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level-raising statement.

We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.

Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$ .

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 show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

Let $G$ be an anisotropic semisimple group over a totally real number field $F$ . Suppose that $G$ is compact at all but one infinite place $v_{0}$ . In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$ -almost simple, not split, and has a Cartan involution defined over $F$ . If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$ , we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $p$ -adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic  $p$ . We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $C$ .

Let  $F$ be a non-archimedean local field of residual characteristic  $p$ , $\ell \neq p$ be a prime number, and  $\text{W}_{F}$ the Weil group of  $F$ . We classify equivalence classes of  $\text{W}_{F}$ -semisimple Deligne  $\ell$ -modular representations of  $\text{W}_{F}$ in terms of irreducible  $\ell$ -modular representations of  $\text{W}_{F}$ , and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the  $\ell$ -modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

We study genuine local Hecke algebras of the Iwahori type of the double cover of $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ and translate the generators and relations to classical operators on the space $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$ , $M$ odd and square-free. In [9] Manickam, Ramakrishnan, and Vasudevan defined the new space of $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$ that maps Hecke isomorphically onto the space of newforms of $S_{2k}(\unicode[STIX]{x1D6E4}_{0}(2M))$ . We characterize this newspace as a common $-1$ -eigenspace of a certain pair of conjugate operators that come from local Hecke algebras. We use the classical Hecke operators and relations that we obtain to give a new proof of the results in [9] and to prove our characterization result.

We study $\text{Sp}_{2n}(F)$ -distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$ -adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$ -packets with no distinguished members that transfer under base change to $\text{Sp}_{2n}(E)$ -distinguished representations of $\text{GL}_{2n}(E)$ .

We characterize the cuspidal representations of $G_{2}$ whose standard ${\mathcal{L}}$ -function admits a pole at $s=2$ as the image of the Rallis–Schiffmann lift for the commuting pair ( $\widetilde{\text{SL}}_{2}$ , $G_{2}$ ) in $\widetilde{\text{Sp}}_{14}$ . The image consists of non-tempered representations. The main tool is the recent construction, by the second author, of a family of Rankin–Selberg integrals representing the standard ${\mathcal{L}}$ -function.

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of $\text{GL}(n)$ over a non-archimedean local field by means of the local Jacquet–Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.

We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ to a simpler function of $k$ and $N$ , showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ .

It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found not to be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$ , then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.