Download PDF by Jacquet H., Langlands R.P.: Automofphic forms on GL(2)

By Jacquet H., Langlands R.P.

Show description

Read or Download Automofphic forms on GL(2) PDF

Similar algebra books

Download PDF by Gerd Fischer: Lernbuch Lineare Algebra und Analytische Geometrie: Das

Diese ganz neuartig konzipierte Einführung in die Lineare Algebra und Analytische Geometrie für Studierende der Mathematik im ersten Studienjahr ist genau auf den Bachelorstudiengang Mathematik zugeschnitten. Die Stoffauswahl mit vielen anschaulichen Beispielen, sehr ausführlichen Erläuterungen und vielen Abbildungen erleichtert das Lernen und geht auf die Verständnisschwierigkeiten der Studienanfänger ein.

Lie Groups and Lie Algebras II - download pdf or read online

A scientific survey of all of the uncomplicated effects at the idea of discrete subgroups of Lie teams, awarded in a handy shape for clients. The booklet makes the idea obtainable to a large viewers, and should be a typical reference for a few years to come back.

Additional resources for Automofphic forms on GL(2)

Example text

For a given Φ in S(F 2 ) there is an ideal a such that Φ(x, y) = Φ(x, 0) ′ for y in a. If a is the complement of a Φ(x, y) |x|s+1 |y|s d× x d× y is equal to the sum of Φ(x, y) |x|s+1 |y|s d× x d× y a′ F which has no pole at s = 0 and a constant times Φ(x, 0) |x|s dx F |y|s d× y a If a = pn the second integral is equal to |̟|ns (1 − |̟|s )−1 If Φ(x, 0) dx = 0 F the first term, which defines a holomorphic function of s, vanishes at s = 0 and the product has no pole there. If ϕ0 is the characteristic function of OF set Φ(x, y) = ϕ0 (x) − |̟|−1 ϕ0 (̟ −1 x) ϕ0 (y).

1 ν(−1) σ Replacing ρ by ρ−1 ν0−1 we obtain the first part of the proposition. If ρ = ν then δ(ρν −1 ) = 1. Moreover, as is well-known and easily verified, η(ρν −1 , ̟ r ) = 1 if r ≥ −ℓ, η(ρν −1 , ̟ −ℓ−1 ) = |̟|(|̟| − 1)−1 and η(ρν −1 , ̟ r ) = 0 if r ≤ −ℓ − 2. 4) is equal to −∞ z0−p−r Cn+r (ν)Cn+r (ν −1 ν0−1 ). ν0 (−1)δn,pI+(|̟|−1)−1 z0−p+ℓ+1 Cn−ℓ−1 (ν)Cn−ℓ−1 (ν −1 ν0−1 )− r=−ℓ−2 The second part of the proposition follows. 12) (i) For every n, p, ν and ρ Cn (ν)Cp (ρ) = Cp (ρ)Cn (ν) (ii) There is no non-trivial subspace of X invariant under all the operators Cn (ν).

5) we deduce a relation of the form q λi Cn+r−i (ν) Cn+r (ν) = i=1 where r is a fixed integer and n is any integer greater than p. 3 is a consequence of the following more precise lemma. If pm is the conductor of a character ρ we refer to m as the order of ρ. 6 Let m0 be of the order ν0 and let m1 be an integer greater than m0 . Write ν0 in any manner in the form ν0 = ν1−1 ν2−1 where the orders of ν1 and ν2 are strictly less than m1 . If the order m of ρ is large enough C−2m−2ℓ (ρ) = ν2−1 ρ(−1)z0−m−ℓ η(ν1−1 ρ, ̟ −m−ℓ ) η(ν2 ρ−1 , ̟ −m−ℓ ) and Cp (ρ) = 0 if p = −2m − 2ℓ.

Download PDF sample

Automofphic forms on GL(2) by Jacquet H., Langlands R.P.


by George
4.4

Rated 4.38 of 5 – based on 38 votes