By David Ginzburg, Erez Lapid, David Soudry

This ebook is the 1st of 2 volumes, which symbolize prime issues of present study in automorphic varieties and illustration thought of reductive teams over neighborhood fields. Articles during this quantity mostly symbolize worldwide points of automorphic types. one of the themes are the hint formulation; functoriality; representations of reductive teams over neighborhood fields; the relative hint formulation and sessions of automorphic varieties; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions. The articles are written by means of top researchers within the box, and convey the reader, complex graduate scholars and researchers alike, to the frontline of the full of life learn in those deep, very important subject matters. The significant other quantity (""Contemporary arithmetic, quantity 489"") is dedicated to neighborhood points of automorphic types

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Consider the subgroup H = Y j(V2n ) ⊂ U2 +1 and extend ψY to a character ψH of H(A), by requiring ψ j(V (A)) = 1. Then 2n −1 ϕπ (yj(vh))ψH (y)dvdy Iϕψπ (h) = V2n (F )\V2n (A) Y (F )\Y (A) −1 ϕπ (ej(h))ψH (e)de. = H(F )\H(A) 26 ASHER BEN-ARTZI AND DAVID SOUDRY We may assume h = 1 and set I = Iϕψπ (1), for short. Now note that H may be described as follows ⎫ ⎧⎛ ⎞ z ∗ ∗ ∗ ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ I 0 ∗ ∗ ⎬ ⎨⎜ n ⎜ ⎟ ⎜ ⎟ 1 0 ∗ ⎟ ∈ U2 +1 : z ∈ Z −n , H= ⎜ ⎪ ⎪ ⎪ ⎪ ⎝ In ∗ ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ z whence, H contains the following normal subgroup ⎧⎛ ⎫ ⎞ ⎨ I 0 ∗ ⎬ 1 0 ⎠ ∈ U2 +1 .

An )φν (x) −n ∧ t Wi,ν I i . −n Here, Wi,ν vary in a certain ﬁnite set of Whittaker functions in the ψν,Nm Whittaker model of πν . This set depends on Wν ; φν , φν are positive functions in S(Eνn ) and S(M −n (Eν ), respectively, and also depend on Wν . 3) t x ∧ I −n = φ0ν (a1 , . . , an )(φ )0ν (x)Wν0 ∧ t I , −n n where φ0ν , (φ )ν are the characteristic functions of OE and M −n,n (OEν ), respecν tively. 3) are valid for any generic representation of Um (Fν ) (not necessarily a local factor of a cuspidal automorphic representation).

Assume that of the form IndBm m (Fν ) ν is inert in E, and that Eν /Fν is unramiﬁed. 3) tπν = (diag(χ1 , . . , χ[ m2 ] , 1, . . , 1), φE/F ), where χi is the value of χν on the diagonal element of Bm (Fν ), which has ν in the i-th coordinate, ν−1 in the m−i+1-th coordinate, and 1 elsewhere (in the diagonal) (see [T]). Assume, next, that ν splits in E. Then Um (Fν ) ∼ = GLm (Fν ), and we may take Bm (Fν ) as the standard Borel subgroup of GLm (Fν ). 4) tπν = (diag(χ1 , . . , χm ), 1), where χi is the value of χν on the diagonal element, which has ν in the i-th coordinate, and 1 elsewhere (in the diagonal).