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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
discovered by Pollack, we prove that a cuspidal representation of
is a (weak) functorial lift from the exceptional group
if its (partial) Spin L-function has a pole at
be a connected split reductive group over a finite field
a smooth projective geometrically connected curve over
-adic cohomology of stacks of
-shtukas is a generalization of the space of automorphic forms with compact support over the function field of
. 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.
be an anisotropic semisimple group over a totally real number field
. Suppose that
is compact at all but one infinite place
. In addition, suppose that
-almost simple, not split, and has a Cartan involution defined over
is a congruence arithmetic manifold of non-positive curvature associated with
, we prove that there exists a sequence of Laplace eigenfunctions on
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
-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.
is a connected reductive group over a finite extension
is a field of characteristic
. We prove that the group
admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over
be a non-archimedean local field of residual characteristic
$\ell \neq p$
be a prime number, and
the Weil group of
. We classify equivalence classes of
-modular representations of
in terms of irreducible
-modular representations of
, and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the
-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
and translate the generators and relations to classical operators on the space
odd and square-free. In  Manickam, Ramakrishnan, and Vasudevan defined the new space of
that maps Hecke isomorphically onto the space of newforms of
. We characterize this newspace as a common
-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  and to prove our characterization result.
We develop a general procedure to study the combinatorial structure of Arthur packets for
following the works of Mœglin. This will allow us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.
-distinction for representations of the quasi-split unitary group
variables with respect to a quadratic extension
-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
-packets with no distinguished members that transfer under base change to
-distinguished representations of
We characterize the cuspidal representations of
-function admits a pole at
as the image of the Rallis–Schiffmann lift for the commuting pair (
. 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
We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of
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.
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight
to a simpler function of
, showing that the two are equal whenever
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
It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight
would yield a fast test for whether
is squarefree. We also show how to obtain bounds on the possible square divisors of a number
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
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
, then we show how to probabilistically factor
entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.
In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.
Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.
We construct, over any CM field, compatible systems of
-adic Galois representations that appear in the cohomology of algebraic varieties and have (for all
) algebraic monodromy groups equal to the exceptional group of type
We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for
in various aspects. A main tool is Arthur’s invariant trace formula. While Shin [Automorphic Plancherel density theorem, Israel J. Math.192(1) (2012), 83–120] and Shin–Templier [Sato–Tate theorem for families and low-lying zeros of automorphic
-functions, Invent. Math.203(1) (2016) 1–177] used Euler–Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms
in Theorem 1.1 which have not been studied and a mysterious second term
also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato–Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor
-functions and degree 5 standard
-functions of holomorphic Siegel cusp forms.
In this paper we prove a conjecture relating the Whittaker function of a certain generating function with the Whittaker function of the theta representation
. This enables us to establish that a certain global integral is factorizable and hence deduce the meromorphic continuation of the standard partial
. In fact we prove that this partial
function has at most a simple pole at
is a genuine irreducible cuspidal representation of the group
For the groups
-adic field, we consider the tempered irreducible representations of unipotent reduction. Lusztig has constructed and parametrized these representations. We prove that they satisfy the expected endoscopic identities which determine the parametrization.