By David Ginzburg, Erez Lapid, David Soudry

ISBN-10: 0821847066

ISBN-13: 9780821847060

ISBN-10: 0821847082

ISBN-13: 9780821847084

This booklet is the second one of 2 volumes, which symbolize top issues of present examine in automorphic kinds and illustration conception of reductive teams over neighborhood fields. Articles during this quantity in most cases signify worldwide points 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 varieties; 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 energetic learn in those deep, important themes. The significant other quantity (""Contemporary arithmetic, quantity 488"") is dedicated to international facets of automorphic kinds

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**Extra info for Automorphic Forms and L-functions II: Local Aspects**

**Example text**

The Fourier expansion of Hχ (s0 ) can be computed explicitly, the Fourier coeﬃcients are (up to a power of π) elements of a cyclotomic ﬁeld. This is enough to prove algebraicity of the values of twisted triple L-functions and to study the Galois behavior of the special values. In Section 6, we ﬁxed an odd prime p and we switched from N to N0 · p where N0 is coprime to p. We only considered cusp forms of level (dividing) N0 · p and we twisted by characters of modulus N0 · pv . We improved Hχ in two ways: for a p-ADIC INTERPOLATION FOR TRIPLE L-FUNCTIONS: ANALYTIC ASPECTS 37 positive integer l, we pass from Hχ (s0 ) (which was of level N02 · p2v ) to a function Hχ (s0 )(l) of level N 2 · p such that f1ρ , f2ρ , f3ρ , Hχ (s0 )(l) Γ0 (N02 ·p) ∼ LN0 ·p · L(N0 ·p) (f1ρ ⊗ f2ρ ⊗ f3ρ , s0 , χ) for all Hecke eigenforms fi ∈ S k (N0 · p, ψi ) and all characters χ mod N0 · pv .

The level of these threefold modular forms grows with the conductor of the characters involved. The aim of this section is to control the level. Our calculation can be done more generally; we stick here to the case needed for p -adic interpolation. 1. Exterior twist and trace. We ﬁx the following abstract situation (slightly diﬀerent from previous sections): We start from the data • a ﬁxed number N ; • a prime p with p N ; • three Dirichlet characters ψi mod N ; • a positive integer n; • a character χ1 mod N pn .

Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc. 31 (1975), 79-98. G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783-804. G. Shimura, On the periods of modular forms, Math. Ann. 29 (1977), 211-221. G. Shimura, Conﬂuent hypergeometric functions on tube domains, Math. Ann. 260 (1982), 269-302.

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