By David Ginzburg, Erez Lapid, David Soudry
This booklet is the 1st of 2 volumes, which signify prime subject matters of present study in automorphic kinds and illustration thought of reductive teams over neighborhood fields. Articles during this quantity almost always signify worldwide facets of automorphic varieties. one of the themes are the hint formulation; functoriality; representations of reductive teams over neighborhood fields; the relative hint formulation and classes of automorphic kinds; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions. The articles are written via top researchers within the box, and convey the reader, complicated graduate scholars and researchers alike, to the frontline of the energetic study in those deep, important themes. The spouse quantity (""Contemporary arithmetic, quantity 489"") is dedicated to neighborhood facets of automorphic varieties
Read Online or Download Automorphic Forms and L-functions I. Global Aspects: Israel Mathematical Conference Proceedings : A Workshop in Honor of Steve Gelbart on the Occasion ... 2006 Rehovot and PDF
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Additional info for Automorphic Forms and L-functions I. Global Aspects: Israel Mathematical Conference Proceedings : A Workshop in Honor of Steve Gelbart on the Occasion ... 2006 Rehovot and
Then the integral L(ϕπ , φ, ξτ,s ) converges absolutely and uniformly in vertical strips in the complex plane, away from poles of the Eisenstein series. The integral is Eulerian, for decomposable data, and, given a complex number s0 , there exist data, such that L(ϕπ , ξτ,s ) is equal, up to a holomorphic function, which is non-zero at s0 , to LS (π × (τ ⊗ µ), s) . LS (τ, A, 2s) The L-function LS (π × τ, s) is already known to have an integral representation, namely, by the Langlands-Shahidi method.
This is probably a necessary condition for a proper understanding of the zeta functions and cohomology of Shimura varieties. The argument is long. However, it also seems to be very natural. Here are some fundamental properties of representations that must be brought to bear on the identities (i) and (ii). REPORT ON THE TRACE FORMULA 9 (1) The classiﬁcation of isobaric representations of GL(N ) (Jacquet-Shalika), which generalizes the theorem of strong multiplicity one. (2) The classiﬁcation of automorphic representations that occur in the spectral decomposition of GL(N ) (Mœglin-Waldspurger).
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 . C= ⎝ ⎩ ⎭ I Moreover, ψ C(A) = 1, whence, −1 ψH (e) I= C(A)H(F )\H(A) −1 ψH (e)ϕC π (e)de, ϕπ (ce)dcde = C(F )\C(A) C(A)H(F )\H(A) where we let ϕC π (g) = ϕπ (cg)dc . C(F )\C(A) We will use the following Shalika expansion [Sk], expressing ϕC π in terms of the Whittaker coeﬃcient Wϕψπ (d∧ g) .