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

By John Coates, R. Sujatha

Written through prime staff within the box, this short yet based ebook offers in complete aspect the best evidence 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 type, with sufficient clarification aiding the reader orientate in the course of technical details." --ZENTRALBLATT MATH

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Let a and b be integers which are prime to p and define c(a, b) = (cn (a, b)) by cn (a, b) = −a/2 − ζn −b/2 − ζn ζn ζn a/2 b/2 . 14) We have already remarked that c(a, b) belong to U∞ . Let ζ (s) denote the complex Riemann zeta function. 3. We have (i) δk (c(a, b)) = 0 for k = 1, 3, 5 · · · , (ii) δk (c(a, b)) = (bk − ak )ζζ (1 − k) for k = 2, 4, 6, · · · . 30 2 Local Units Proof. Put f (T ) = (1 + T )−a/2 − (1 + T )a/2 (1 + T )−b/2 − (1 + T )b/2 so that f (πn ) = cn (a, b) for all n ≥ 0. We make the change of variable d T = ez − 1 so that D = dz .

2) Proof. 3) and write Gn = Gal(Fn /Q), Gn = Gal(Fn /Q). Let πn : Zp [Gn ] −→ Zp [Gn ] denote the natural surjection. We claim that πn induces an isomorphism from Zp [Gn ]+ onto Zp [Gn ]. Indeed, it is clear that πn is surjective, and that it maps Zp [Gn ]− to zero. To complete the proof, we note that the Zp -rank of Zp [Gn ]+ is equal to ((p − 1)/2)pn , because Zp [Gn ](i) Zp [Gn ]+ = i even i mod p−1 where the sum on the right is taken over the eigenspaces for the even powers modulo (p − 1) of the character giving the action of G0 on µp .

We can then define − 1)λ) , ν(g) − 1 G where g is any element of G with ν(g) = 1. This is independent of the choice of g because, as remarked earlier ν extends to a ring homomorphism from Λ(G) to Cp . 3 The Mahler Transform The key tool for relating the ring R = Zp [[T ]] of formal power series studied in the previous chapter, to the Iwasawa algebra of the Galois group G = Gal(K∞ /Qp ) is provided by the following remarkable theorem of Mahler [M], whose proof we omit. As usual we define nx to be 1 if n = 0, and x n = x(x − 1) · · · (x − n + 1) n!

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