Cyclotomic Fields and Zeta Values by John Coates, R. Sujatha

By John Coates, R. Sujatha

Written by means of prime employees within the box, this short yet stylish ebook offers in complete element the easiest facts of the "main conjecture" for cyclotomic fields. Its motivation stems not just from the inherent great thing about the topic, but in addition from the broader mathematics curiosity of those questions.
From the experiences: "The textual content is written in a transparent and tasty variety, with adequate clarification supporting the reader orientate in the middle of technical details." --ZENTRALBLATT MATH

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We denote by Un1 the subgroup {x ∈ Un : x ≡ 1 mod pn }. More generally, if Z is any subgroup of Un , we write Z 1 = Z ∩ Un1 . 4 Iwasawa’s Theorem 53 Note that the index of Z 1 in Z always divides p − 1, because Z 1 is the kernel of the reduction map from Z into F× p . In particular, this index 1 1 is prime to p. Also, note that Un and Un are now Zp -modules, whereas this is plainly not true for Un and Un themselves. 9) taken with respect to the norm maps. Since Un1 and Cn1 are compact 1 and C 1 .

12). 2. For all k ≥ 1, and all u in U∞ , we have G ˜ χ(g)k dL(u) = (1 − pk−1 )δk (u). 12) Proof. 10) gives G χ(g)k dλ = Z× p xk d(χ(λ)). ˜ Also, via our identification of Λ(Z× p ) with a subset of Λ(Zp ), the integral above on the right has the same value if we integrate over the whole ˜ ˜ of Zp . Now take λ = L(u), so that, by definition, we have χ( ˜ L(u)) = Υ(L(fu )), where we recall that L(fu )(T ) = fu (T )p 1 . 12) is equal to Dk−1 (hu (T ) − ϕ(hu )(T )) T =0 , where hu (T ) = (1 + T ) fu (T ) .

Let f be in W . Recalling that ϕ(f )(T ) = f ((1 + T )p − 1) and applying ∆ to the equation f (ξ(1 + T ) − 1), ϕ(f ) = ξ∈µp we obtain immediately that ψ(∆(f )) = ∆(f ). The final assertion of the lemma is obvious. In fact, the following stronger result is true, but its proof is subtle and non-trivial. 6. We have ∆(W ) = Rψ=1 . The strategy of the proof is to use reduction modulo p. 9) and let x → x be the reduction map. If Y is any subset of R, then we denote by Y its image in Ω under the reduction map.

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