The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between (stable) trace formulas of different groups G1,G2 for every morphism G1L→LG2 their L-groups. This conjecture can be seen a special case more general which replaces reductive by spherical varieties and formula its generalization, relative formula. goal this article continuation...

We prove in this note that the global geometric theta lifting for the pair (H,G) is compatible with the Whittaker normalization, where (H,G) = (SO2n, Sp2n), (Sp2n, SO2n+2), or (GLn, GLn+1). More precisely, let k be an algebraically closed field of characteristic p > 2. Let X be a smooth projective connected curve over k. For a stack S write D(S) for the derived category of étale constructible Q̄...

1.0 This paper, which is a sequel to [6], is a step towards a geometric version of the Howe correspondence (an analogue of the theta-lifting in the framework of the geometric Langlands program). We consider only the (unramified) dual reductive pair (H = GO2m, G = GSp2n) over a smooth projective connected curve X. Let BunG (resp., BunH) denote the stack of Gtorsors (resp., H-torsors) on X. Using...

Introduction 1 1. The unramified global Langlands correspondence 5 2. Classical local Langlands correspondence 9 3. Geometric local Langlands correspondence over C 12 4. Center and opers 18 5. Opers vs. local systems 23 6. Harish–Chandra categories 26 7. Local Langlands correspondence: unramified case 30 8. Local Langlands correspondence: tamely ramified case 41 9. Ramified global Langlands cor...

Suppose E/F is a quadratic extension of number fields and GU(2, 2) is the quasi-split unitary similitude group attached to E/F . We prove functoriality from globally generic cuspidal representations of GU(2, 2)(AF ) to automorphic representations of GL6(AF ) corresponding to the twisted exterior square map on the dual side. For a split quadratic algebra E/F , the twisted exterior square map red...

We report briefly on the present state of the trace formula and some of its applications. This article is a summary of the two hour presentation/discussion on the trace formula. The proposed topic was very broad. It included a recapitulation of the trace formula, past and present, as well as an outlook for its future. The article will treat these matters in only the most concise terms. I includ...

Let f(z) = ∑∞ n=1 af (n)q n ∈ S k (Γ0(N)) be a newform with squarefree level N that does not have complex multiplication. For a prime p, define θp ∈ [0, π] to be the angle for which af (p) = 2p (k−1)/2 cos θp. Let I ⊂ [0, π] be a closed subinterval, and let dμST = 2 π sin 2 θdθ be the Sato-Tate measure of I. Assuming that the symmetric power L-functions of f satisfy certain analytic properties ...

The Langlands program of number theory, or what we might call Langlands duality, was proposed in more or less its present form by Robert Langlands, in the late 1960s. It is a kind of unified scheme for many results in number theory ranging from quadratic reciprocity, which is hundreds of years old, to modern results such as Andrew Wiles’ proof of Fermat’s last theorem, which involved a sort of ...

For a split reductive group $G$ over number field $k$, let $\rho$ be an $n$-dimensional complex representation of its dual $G^\vee(\mathbb{C})$. any irreducible cuspidal automorphic $\sigma$ $G(\mathbb{A})$, where $\mathbb{A}$ is the ring adeles in \cite{JL21}, authors introduce $(\sigma,\rho)$-Schwartz space $\mathcal{S}_{\sigma,\rho}(\mathbb{A}^\times)$ and $(\sigma,\rho)$-Fourier operator $\...