By John Coates, R. Sujatha
Written by way of best employees within the box, this short yet dependent ebook provides in complete aspect the easiest facts of the "main conjecture" for cyclotomic fields. Its motivation stems not just from the inherent fantastic thing about the topic, but additionally from the broader mathematics curiosity of those questions. From the reports: "The textual content is written in a transparent and tasty kind, with adequate rationalization supporting the reader orientate in the middle of technical details." --ZENTRALBLATT MATH
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Additional info for Cyclotomic Fields and Zeta Values
Write Step(G) for the sub-algebra of locally constant functions, which is easily seen to be everywhere dense. We now explain how to integrate any continuous Cp -valued function on G against an element λ of Λ(G). We begin with locally constant functions. Suppose that f in Step(G) is locally constant modulo the subgroup H of G. 1) x∈G/H where the cH(x) lie in Zp . We then deﬁne f dλ = G cH(x)f (x). x∈G/H One sees easily that this is independent of the choice of H. Since the cH(x) lie in Zp , we have ≤ f .
1. We have (1 − ϕ)R = T R. Proof. The inclusion of (1 − ϕ)R in T R is plain. Conversely, if h is any element of T R, let us show that it lies in (1 − ϕ)R. For each n ≥ 0, n we deﬁne ωn (T ) = (1 + T )p − 1. 4 The Logarithmic Derivative 21 where hn is a polynomial in Zp [T ] of degree less than pn , and rn is an element of R. Deﬁne n−1 ϕi (hn−i ). ln = i=0 Clearly, we have ln+1 − ϕ(ln ) = hn+1 . Since hn+1 converges to h in R, it suﬃces to show that ln converges to some l in R, because then we would have h = (1 − ϕ)l.
7, there exists a unique u = (un ) in U∞ such that f = fu and hence we have fu ((1 + T )p − 1) = fu (T )p . This implies that upn = un−1 for all n ≥ 1 and that fu (0) is in µp−1 . But then fu (0) = 1 since fu ≡ 1 mod p and so (un ) ∈ Tp (µ). Thus there exists a in Zp such that u = (ζn )a , whence f (T ) = (1 + T )a . This proves that ker(L) = A. It is clear that α ◦ L = 0, and the surjectivity of α follows from noting that ψ(1 + T ) = 0 and that α(1 + T ) = 1. Hence it only remains to prove that ker(α) ⊂ Im(L), which is the delicate part of the proof of the theorem.
Cyclotomic Fields and Zeta Values by John Coates, R. Sujatha