Armand Borel's Automorphic Forms, Representations, and L-Functions: PDF

By Armand Borel

Includes sections on Reductive teams, representations, Automorphic kinds and representations

Show description

Read Online or Download Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1 PDF

Best number theory books

New PDF release: The Prime Numbers and Their Distribution (Student

We've been serious about numbers--and best numbers--since antiquity. One impressive new path this century within the learn of primes has been the inflow of rules from chance. The target of this e-book is to supply insights into the top numbers and to explain how a chain so tautly made up our minds can include any such amazing quantity of randomness.

Download e-book for kindle: Mathematical Modeling for the Life Sciences by Jacques Istas

Presenting quite a lot of mathematical types which are presently utilized in existence sciences might be considered as a problem, and that's exactly the problem that this e-book takes up. after all this panoramic examine doesn't declare to supply a close and exhaustive view of the various interactions among mathematical versions and existence sciences.

Download e-book for kindle: The Theory of Algebraic Number Fields by David Hilbert

This ebook is a translation into English of Hilbert's "Theorie der algebraischen Zahlkrper" most sensible referred to as the "Zahlbericht", first released in 1897, within which he supplied an elegantly built-in review of the advance of algebraic quantity concept as much as the tip of the 19th century. The Zahlbericht supplied additionally an organization origin for additional examine within the topic.

Additional resources for Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1

Sample text

We shall often need the following elementary lemma, which enables us to identify Λ(G) with a sub-algebra of Λ(G). Let J = {1, ι} = Gal(F∞ /F∞ ). If M is any Zp [J ]-module, then since p is odd, there is the decomposition M = M + ⊕ M − , where M + = 1+ι 1−ι M, M − = M. 2 2 In particular, we have Λ(G) = Λ(G)+ ⊕ Λ(G)− . 1. The restriction to Λ(G)+ of the natural surjection from Λ(G) onto Λ(G) induces an isomorphism Λ(G)+ Λ(G). 2) Proof. 3) and write Gn = Gal(Fn /Q), Gn = Gal(Fn /Q). Let πn : Zp [Gn ] −→ Zp [Gn ] denote the natural surjection.

We note that, since the norm map from Kn to Kn−1 induces the identity map on the residue fields, we have 1 1 U∞ = µp−1 × U∞ , U∞ = µp−1 × U∞ . 11) can be rewritten as an exact sequence L1 1 −→ Λ(G) −→ Tp (µ) −→ 0, 0 −→ Tp (µ) −→ U∞ 1 . As p is odd, the above sequence with L1 being the restriction of L to U∞ remains exact after taking invariants under J = {1, ι}. Since Tp (µ)J = 0, there is a canonical Λ(G)-isomorphism 1 L1 : U∞ Λ(G). b, C∞ so that 1 ) = Λ(G)L1 (b). 4 shows that L1 (b) = ζ p θ+ (e, 1) where θ+ (e, 1) denotes the image of θ(e, 1) in Λ(G).

6), to conclude that there exists w in W with ∆(w) = h. By construction and the commutativity of the diagram, we have g = D(f ) = DL(w). Hence f = L(w) by the injectivity of D. Thus f belongs to the image of L and the proof of the theorem is complete. 2 to define the higher logarithmic derivatives of elements of U∞ . We study these maps and show by a mysterious, but elementary, calculation going back to Kummer, that the values of the Riemann zeta function at the odd negative integers arise as the higher logarithmic derivatives of cyclotomic units.

Download PDF sample

Rated 4.00 of 5 – based on 39 votes