# Download Automorphic Forms and L-functions II: Local Aspects by David Ginzburg, Erez Lapid, David Soudry PDF

By David Ginzburg, Erez Lapid, David Soudry

This e-book is the second one of 2 volumes, which symbolize prime subject matters of present examine in automorphic varieties and illustration idea of reductive teams over neighborhood fields. Articles during this quantity generally symbolize international features of automorphic varieties. one of the subject matters 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 through top researchers within the box, and convey the reader, complex graduate scholars and researchers alike, to the frontline of the lively study in those deep, very important themes. The significant other quantity (""Contemporary arithmetic, quantity 488"") is dedicated to worldwide facets of automorphic kinds

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**Automorphic Forms and L-functions II: Local Aspects**

This e-book is the second one of 2 volumes, which symbolize major subject matters of present learn in automorphic varieties and illustration idea of reductive teams over neighborhood fields. Articles during this quantity regularly characterize worldwide features of automorphic varieties. 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.

**Additional resources for Automorphic Forms and L-functions II: Local Aspects**

**Example text**

Let p, N ψi , χi , φ be as before and let F be a C ∞ modular form on H3 of weight m, level Γ0 (N pn ) and nebentypus φ with Fourier expansion (Z = X + iY ∈ H3 ). a(T, Y )e2πi·tr(T Z) F (Z) = T Then the function 13 0 χ1 (α)χ2 (β)χ3 (γ)F |m Fˆ := ri mod p2n−1 α,β,γ has a Fourier expansion in the “cusps” u N2 Fˆ |m u v N2 p = 13 N 2 pR 03 13 of the following form ⊗3 v p S(α, β, γ) 13 1 2πi·tr( p2n T ·Z) b(T, Y )e T tii ≡0(p2n−1 ) with b(T, Y ) = a T, 1 Y p2n × φN (p)(χ1N χ2N χ3N )(p2 )φp (N )6 × (χ1N χ2N χ3N )(p3n )GN (t12 , χ1N )GN (t13 , χ3N GN (t23 , χ3N ) 1 × 3nm χ1p χ2p χ3p (−2N )Gpn (t12 , χ1p φp ) · Gpn (t13 , χ2p φp )· p 1 × 3nm χ1p χ2p χ3p (−2N )Gpn (t23 , χ3p φp ).

If we prefer, we can express this in terms of modular forms for Γ0 (N 2 , p), comparing L(Fˆ ω ⊗3 ), f1 ω, f2 ω, f3 ω Γ (N 2 ,p) m with k1 L(Fˆ ) ω(U 0 (pl ))⊗3 , f1 k2 k1 ω, f2 0 k3 k2 ω, f3 k3 ω Γ0 (N 2 ,p) . As a main result of this section, we obtain the Fourier expansion of a level N 2 p modular form of three variables (obtained from a modular form F on H3 ) in an appropriate cusp: For simplicity, we state it only for a holomorphic modular form F : Proposition. Suppose that the data N, p, ψi , n, χi , φ are given as at the begin3 ning of this section.

1) in terms of these data, using • c˜1 c˜2 a ˆ ≡ −b1 − b2 mod N N N c˜1 c˜2 1 c 4 − c3 ≡ det C N N ˜ 1 det(C) ≡ N 2 det(C) 1 c γ ≡ det C N N ˆ · b ≡ det(C) Nc • t≡a γ 1 1 c γ N det(C) dδ (−2 N N uv ) ≡ −2uv 1 dδ . 3) N k+2s ψf ψf χ1 (−v)χ1 χ2 (u)χ2 χ3 ( ) N N N N u,v,c,d,γδ, f ρ R w1 v 2 + S w 4 u2 − j(R, w1 )−k j(S, w4 )−k 2uv dδ N (R w1 ) (S w4 ) (R w1 )v 2 + (S w4 )u2 s (vX1 − uX2 )b . In the calculations above, we ignored to some extent questions of dependence on choosing representatives. 3) is perfectly independent of any kind of choice of representatives !