On the cyclotomic main conjecture for the prime 2

Abstract

Let l be prime number, m 0 an integer prime to l andthe cyclotomic Iwasawa algebra of "tame level m 0 ". Usingétale cohomology one can define a certain perfect complex of Λ-modules ∆ ∞ (see section 1.2 below) and a certain basis L of the invertible In this article we give a proof of Theorem 1.2 for l = 2. This was claimed as Theorem 5.2 in the survey paper [9] but the proof given there, arguing separately for each height one prime q of Λ, is incomplete at primes q which contain l = 2. The argument given in [9][p.95] is an attempt to use only knowledge of the cohomology as well as perfectness of the complex ∆ ∞ q but it turns out that this information is insufficient. Here we shall use the techniques of the paper [6], such as the Coleman homomorphism and Leopoldt's result on the Galois structure of cyclotomic integer rings, to construct the complex ∆ ∞ q more explicitly and thereby verify Theorem 1.2.What is peculiar to the situation l = 2 is not only that Λ is never regular (due to the presence of the complex conjugation) but also that the l-adic L-function L is quite differently defined for even and odd characters. For even characters one interpolates first derivatives of Dirichlet L-functions via cyclotomic units, for odd characters one interpolates the values of these functions via Stickelberger elements. A proof of Theorem 1.2 therefore in some sense involves a "mod 2 congruence" between Stickelberger elements and cyclotomic units. Given the explicit nature of both objects it is perhaps not surprising that this congruence turns out to be an elementary statement which is arrived at, however, only after some rather arduous computations. The statement is the following. Let M ≡ 1 mod 4 be an integer,

On the p‐adic Stark conjecture at s=1 and applications

“…Proof The first claim is [15, Corollary 1.3], which crucially depends on the results of [26, 27, 45]. The second claim is well known to experts; we give a proof here for the convenience of the reader.…”

“…Remark A more general version of Corollary 8.7 (without the restrictions that the fields in question are totally real or that is odd) is certainly well known, but our approach provides a new proof in this particular setting. The method of Burns and Flach [26] uses the validity of (as proven outside the 2‐primary part by Burns and Greither [27] and at by Flach [45]) and compatibility with the functional equation. In some respects, our approach is closer to the proof of Huber and Kings [51] of the Bloch–Kato conjecture for Dirichlet characters, which implies (among other results) for odd primes ; of course, this is somewhat weaker than .…”

“…The main unconditional results to date on the above special cases of the ETNC are as follows (by the above discussion most of these can also be phrased in terms of the LTCs). Burns and Greither [27] showed that if is a finite abelian extension and with , then the ETNC holds for the pair ; Flach [45] resolved important technical difficulties at the prime , allowing in the above to be replaced with . Moreover, Burns and Flach [26] showed that the analogous results hold when .…”

The Equivalence of Rubin's Conjecture and the Etnc/LRNC for Certain Biquadratic Extensions

“…9.2], Z(L/K) implies Rubin's Conjecture for all Rubin data (S, T, r) for L/K. Therefore, cases (i) and (ii) above follow from the truth of Z(L/K) for L/K in the specified situations: (i) by Burns-Greither [7] and Flach [12], and (ii) by Bley [1].…”

“…This is important, for if L/Q were abelian, then Z(L/K) would follow from the main results of [7] and [12].…”