By John Coates, R. Sujatha

Written through best employees within the box, this short yet based booklet provides in complete aspect the easiest facts of the "main conjecture" for cyclotomic fields. Its motivation stems not just from the inherent great thing about the topic, but additionally from the broader mathematics curiosity of those questions.

From the stories: "The textual content is written in a transparent and engaging type, with adequate clarification aiding the reader orientate in the course of technical details." --ZENTRALBLATT MATH

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**Extra resources for Cyclotomic Fields and Zeta Values**

**Example 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.