By David Ginzburg, Erez Lapid, David Soudry

ISBN-10: 0821847066

ISBN-13: 9780821847060

ISBN-10: 0821847082

ISBN-13: 9780821847084

This e-book is the second one of 2 volumes, which symbolize major issues of present learn in automorphic kinds and illustration concept of reductive teams over neighborhood fields. Articles during this quantity almost always signify international 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 classes of automorphic varieties; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions. The articles are written by means of prime researchers within the box, and produce the reader, complex graduate scholars and researchers alike, to the frontline of the lively learn in those deep, very important issues. The spouse quantity (""Contemporary arithmetic, quantity 488"") is dedicated to worldwide features of automorphic kinds

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

**Example text**

Then pt (s) is exactly the shift which occurs in the triple L-function on the right side of our Theorem . Then we consider K1 := = k2 + k3 − k1 −1 2 k2 + k3 − k1 k2 + k3 − 1 − , . . , k2 + k3 − 1 2 pt (0) | 0 ≤ t ≤ and K2 := = k2 + k3 − k1 −1 2 k1 + k2 − k3 k1 , . . , k1 + −1 . 2 pt (2 − m) | 0 ≤ t ≤ Then the union of these sets is exactly the set of critical values. The smallest integer in K1 equals the largest number in K2 iﬀ there is a central critical value (this corresponds to the case m = 2); otherwise these two sets are disjoint.

102 (1967), 30-43. [Kl90] H. Klingen, Introductory Lectures on Siegel Modular Forms, Cambridge University Press, 1990. [Kit84] Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J. 95 (1984), 73-84. [Ku-Le] T. -W. Leopoldt, Eine p-adische Theorie der Zetawerte, J. reine angew. Math. 214/215 (1964), 328-339. [La] S. Lang, Introduction to Modular Forms, Springer Verlag, 1976. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285-315. [Ma71] H. Maaß, Siegel’s Modular Forms and Dirichlet Series, Lecture Notes in Math.

7]. , not on φ). Remark. 3. 1 πi 3t+λ+ν D0 (m, k + b, k + λ, k + ν) Conclusion. From the statements above, we get in a standard way that for s0 ∈ Zcrit we have an algebraicity result LS (f1 ⊗ f2 ⊗3 , s0 , χ) k +k (2πi) 1 2 +k3 −3 f1 , f1 f2 , f2 f3 , f3 ∈ Q, where , denotes the Petersson scalar product; we can even get a precise formula for the action of Aut(C). 6. Trace computations In the previous section we constructed holomorphic modular forms in three variables whose threefold Petersson product against f, g, h gives special values of the triple L-functions.

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

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