Shalika periods on GL2(D) and GL4

Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

Transfer operators and Hankel transforms between relative trace formulas, I: Character theory

We study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

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

Twisted Linear Periods and a New Relative Trace Formula

We elaborate an explicit version of the relative trace formula on ${\rm PGL}(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution resul

On the relative trace formula

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

Our aim is to describe the image and fibres of the base change lift constructed by Langlands, for solvable extensions of number fields. A particular instance of the principle of functoriality enunciated by Lang- lands, is the base change map relating automorphic forms on GL2(Ak) to forms on GL2(AK), for an extension K/k of number fields. Using the trace formula and following earlier workby Saito and Shintani, Langlands in (L), established the existence of the base change lift BCK/k sending cuspidal automorphic rep- resentations of GL2(Ak) to automorphic representations of GL2(AK), provided K/k is a cyclic extension of prime degree. Langlands further characterized the image and the fibres of the base change map BCK/k. Given an automorphism σ of K, and an automorphic representation Π of GL2(AK), let Π σ denote the automorphic representation of GL2(AK) defined by Π σ (g )=Π (σ −1 (g)). The descent property for invariant, automorphic representations established by Langlands is that if Π is a cuspidal automorphic representation of GL2(AK), which is invariant with respect to the action of the Galois group G(K/k )o f a cyclic extension K/k of prime degree, then Π lies in the image of the base change map BCK/k from automorphic representations on GL2(Ak) to automor- phic representations on GL2(AK). Further if π1 and π2 are distinct, cuspidal automorphic representations of GLn(Ak) which base change to Π, then

Local Orbital Integrals of a Relative Trace Formula

Let G=mathrm{GL}_{2n} over a totally real number field F and nge 2. Let Pi be a cuspidal automorphic representation of G(mathbb {A}), which is cohomological and a functorial lift from SO(2n+1). The latter condition can be equivalently reformulated that the exterior square L-function of Pi has a pole at s=1. In this paper, we prove a rationality result for the residue of the exterior square L-function at s=1 and also for the holomorphic value of the symmetric square L-function at s=1 attached to Pi . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm {GL}_2$ . Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$ , eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level- $1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$ .The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

In this paper we prove the functoriality of the exterior square of cusp forms on G L 4 GL_{4} as automorphic forms on G L 6 GL_{6} and the symmetric fourth of cusp forms on G L 2 GL_{2} as automorphic forms on G L 5 GL_{5} . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain L L -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of G L 4 GL_{4} and the absolute convergence of the exterior square L L -functions of G L 4 GL_{4} . Second, we prove that the fourth symmetric power L L -functions of cuspidal representations of G L 2 GL_{2} are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound 3 26 \frac {3}{26} for Hecke eigenvalues of Maass forms over any number field.

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.