The Langlands Functoriality Conjecture is the number theorist’s “grand unified theory". It describes and elucidates the platonic realm of modular forms, L-functions, and motives—a world arguably no less real, in its richness, than the physicist’s universe of elusive particles and far-flung galaxies. The timely volume under review (referred to henceforth as CKM) is made up of three distinct cont...

We give a short introduction to Beyond Endoscopy, a proposal by Langlands for attacking the general principle of functoriality. We shall try to motivate the proposal by emphasizing its structural similarities with the actual theory of endoscopy. We shall then discuss a few of the many problems that will need to be solved, some of which are suggested by the recent work of A. Altuğ . Mathematics ...

Let k be an algebraically closed field of characteristic > 2, F = k((t)) and G = Sp 2d . In this paper we propose a geometric analog of the Weil representation of the metaplectic group G̃(F ). This is a category of perverse sheaves on some stack, on which G̃(F ) acts by functors. This construction will be used in [5] (and subsequent publications) for the proof of the geometric Langlands functoria...

The Selberg-Arthur trace formula is one of the tools available for approaching the conjecture of global functoriality in the Langlands program. Global functoriality is described within this volume in [Kn2]. We start with reductive groups G and H, say over the rationals Q for simplicity. We assume that G is quasisplit, and we suppose that we are given an L homomorphism ψ : H → G. From an automor...

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = GO2m, G = GSp2n over X , where H splits over an étale two-sheeted covering π : X̃ → X . Write BunG and BunH for the stacks of G-torsors and H-torsors on X . We show that for m ≤ n (respectively, for m > n) the theta-lifting functor FG : D(BunH) → D(BunG) (respectively, FH : D(B...

This article is an introduction to automorphic forms on the adeles of a linear reductive group over a number field. The first half is a summary of aspects of local and global class field theory, with emphasis on the local Weil group, the L functions of Artin and Hecke, and the role of Artin reciprocity in relating the two kinds of L functions. The first half serves as background for the second ...

Let F be a p-adic field of characteristic 0, and let A be an F -central division algebra of dimension dA over F . In the paper we first develop the representation theory of GL(m,A), assuming that holds the conjecture which claims that unitary parabolic induction is irreducible for GL(m,A)’s. Among others, we obtain the formula for characters of irreducible unitary representations of GL(m,A) in ...

We generalize a beautiful method of Blasius and Ramakrishnan, that in order to exhibit particular instances of the Langlands functorial correspondence, it is enough to show that the correspondence holds in the semistable case, provided the arithmetic data we are considering is closed with respect to cyclic base change and descent.

‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

1.1. Earlier work. Let ` be a prime. In our previous work [KLS], which generalised a result of Wiese [W], the Langlands functoriality principle was used to show that for every positive integer t there exists a positive integer k divisible by t such that the finite simple group Cn(`) = PSp2n(F`k) is the Galois group of an extension of Q unramified outside {∞, `, q} where q 6= 2 is a prime that d...