The equivariant Tamagawa number conjecture: a survey

Flach, Matthias; Greither, Cornélius

doi:10.1090/conm/358/06537

Cited by 44 publications

(47 citation statements)

References 53 publications

(52 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is a special case of very general conjectures on motivic L-values put forward by Kato [14] and Kato and Fukaya [10]. Theorem 1.2 for l = 2 not only confirms Kato's point of view of L-values and l-adic L-functions as bases of determinant line bundles quite beautifully but also has other number theoretic consequences which were already noted in [9][5.1] and [6][Cor. 1.2-1.4].…”

supporting

confidence: 67%

“…The proof of Theorem 1.2 for l = 2 3.1. Recollections from [9]. In this section we continue the proof of Theorem 1.2 for l = 2 where we have left it off in [9].…”

Section: Remark Note That the Elementmentioning

confidence: 82%

“…The compact support etale cohomology is defined as in [5][p. 522]. We recall the computation of the cohomology of ∆ ∞ from [9]. We have H i (∆ ∞ ) = 0 for i = 1, 2, a canonical isomorphism…”

Section: Global Iwasawa Theorymentioning

confidence: 99%

“…For odd primes l this theorem is essentially due to Burns and Greither [7] and a proof of this precise statement with this precise notation was given in [9][Sec. 5.1].…”

Section: Global Iwasawa Theorymentioning

confidence: 99%

“…After recalling material from [9] and [6] in section 1 we prove in section 2 a statement which might be regarded as a functional equation of L and which is a refinement of a result going back to Iwasawa. We finally give the proof of Theorem 1.2 in section 3 and collect some technical computations related to the Shapiro Lemma in an Appendix.…”

mentioning

confidence: 94%

See 4 more Smart Citations

On the cyclotomic main conjecture for the prime 2

Flach¹

2011

Journal für die reine und angewandte Mathematik (Crelles Journa

Self Cite

View full text Add to dashboard Cite

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,

show abstract

supporting

confidence: 67%

“…The proof of Theorem 1.2 for l = 2 3.1. Recollections from [9]. In this section we continue the proof of Theorem 1.2 for l = 2 where we have left it off in [9].…”

Section: Remark Note That the Elementmentioning

confidence: 82%

Section: Global Iwasawa Theorymentioning

confidence: 99%

“…For odd primes l this theorem is essentially due to Burns and Greither [7] and a proof of this precise statement with this precise notation was given in [9][Sec. 5.1].…”

Section: Global Iwasawa Theorymentioning

confidence: 99%

mentioning

confidence: 94%

See 3 more Smart Citations

On the cyclotomic main conjecture for the prime 2

Flach¹

2011

Journal für die reine und angewandte Mathematik (Crelles Journa

Self Cite

View full text Add to dashboard Cite

show abstract

Symmetric Powers of Elliptic Curve L-Functions

Martin

Watkins

2006

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The conjectures of Deligne, Beȋlinson, and Bloch-Kato assert that there should be relations between the arithmetic of algebrogeometric objects and the special values of their L-functions. We make a numerical study for symmetric power L-functions of elliptic curves, obtaining data about the validity of their functional equations, frequency of vanishing of central values, and divisibility of Bloch-Kato quotients.

show abstract