By David Ginzburg, Erez Lapid, David Soudry

This ebook is the second one of 2 volumes, which characterize top subject matters of present study in automorphic types and illustration conception of reductive teams over neighborhood fields. Articles during this quantity normally symbolize worldwide features of automorphic kinds. one of the themes are the hint formulation; functoriality; representations of reductive teams over neighborhood fields; the relative hint formulation and sessions of automorphic types; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions. The articles are written via top researchers within the box, and produce the reader, complex graduate scholars and researchers alike, to the frontline of the energetic learn in those deep, important subject matters. The significant other quantity (""Contemporary arithmetic, quantity 488"") is dedicated to international facets of automorphic kinds.

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**Extra resources for Automorphic Forms and L-functions II: Local Aspects (Contemporary Mathematics)**

**Sample text**

6) × GN (ψg ψf χ1 , −2n) n=1 χ1 (−u)χ2 (v)(ψg ψf χ1 )(uv) (u,v)=1 a¯f (n)a¯g (nv 2 )a¯h (nu2 ) . 3. Gauss sums and bad Euler factors. 6) is a new phenomenon (such Gauss sums did not appear in the doubling method [Boe-Schm]). To analyze the role of the Gauss sums, we ﬁrst separate the contributions of “good primes” from those of “bad primes” (in the case of Hecke eigenforms for Hecke operators T (q) for all primes q N ). Then we can use a partial multiplicativity property: For n = n1 · n2 with (n1 , n2 ) = 1 and (n2 , 2N ) = 1, we can write GN (ψg ψf χ1 , −2n) = (ψg ψf χ1 )(n2 )GN (ψg ψf χ1 , −2n1 ) af (n) = λf (n2 )af (n1 ).

Soc. Lect. Note Ser. 235 (1996), 1-46. [BSSP] S. B¨ ocherer, P. Sarnak and R. Schulze-Pillot, Arithmetic and Equidistribution of measures on the sphere, Communications in Math. Physics 242 (2003), 67-80. [BP] S. B¨ ocherer and A. Panchishkin, Admissible measures attached to triple products of elliptic cusp forms, Documenta Mathematica, Extra volume (2006), 77-132 (electronic). [Co] J. Coates, On p-adic L-functions, S´ em. Bourbaki, vol. 1988/89, Asterisque no. 177178 (1989), Exp. no. 701, 33-59.

Ak (T, φ) = γ(k) × q We recall from [Kit84, Kat97] that fq (T, X) ∈ Q[X] is an explicitly given polynomial in X, which is 1 unless q divides det(2T ). The “archimedean factor” γ(k) equals γ(k) = (−1) 3k 2 3 23k−3 π 3k− 2 ∈ π 3k−2 × γ o (k), Γ(k)Γ(k − 12 )Γ(k − 1) where γ o (k) is an elementary rational number (not depending on φ). Therefore we can describe the action of the Galois group ak (T, φ) π 3k−2 σ = ak (T, φσ ) π 3k−2 (σ ∈ Aut(C)). If the Eisenstein series deﬁnes a holomorphic modular form (which is always the case unless k = 3), the Galois action on the modular form is the same as on these Fourier coeﬃcients of maximal rank.