Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

By A. Huber, G. Kings. 2001

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The Tamagawa number conjecture proposed by Bloch and Kato in [BlKa] describes the “special values” of L-functions of motives in terms of cohomological data. Special value means here the leading coefficient of the Taylor series of an L-function at an integral point. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of Galois-modules, more precisely modules for the Iwasawa algebra. We prove: Theorem 4.4.1, 5.1.3, 1.3.1: Let χ be a Dirichlet character. Then the Main Conjecture of Iwasawa theory holds for χ and p 6= 2 and the Bloch-Kato conjecture holds up ot powers of 2 for L(χ, r) with r ∈ Z. Already in the simplest case of the Tamagawa number conjecture – the Riemann ζ-function (see [BlKa] Theorem 6.1) – the Main Conjecture due to Mazur and Wiles is used by Bloch and Kato in an essential way. It was clear to the experts that the Main conjecture for a Dirichlet character and a prime p should be used in order to prove the p-part of the Bloch-Kato conjecture (see e.g. the list in Fontaine’s Bourbaki talk [Fo] § 10 and [PR3]). There are two versions of the Main Conjecture depending on parity. Both are needed in order to deduce the Bloch-Kato conjecture in general. The first version involving the p-adic L-function was proved by Mazur and Wiles. Under the condition that p does not divide Φ(N), the second version involving cyclotomic units follows from the Mazur-Wiles case. Under the same restriction, it was also proved directly by Rubin using Kolyvagin’s Euler systems. The missing cases need to be addressed. What we want to advocate strongly is the insight, due to Kato in [Ka2] and [Ka1] and Perrin-Riou [PR3], that the Bloch-Kato conjecture and the Main Conjecture are two incarnations of the same mathematical content. Knowing the p-part of the Bloch-Kato conjecture for all fields in the cyclotomic tower is equivalent to the Main Conjecture formulated appropriately. Our approach is to take Kato’s viewpoint seriously and to prove the Main Conjecture without any restriction on the prime p from the Bloch-Kato conjecture. As there is one basic case in which the Bloch-Kato conjecture is known – the analytic class number formula for ζF (0) where F is a number

Subfields: Geometry and Topology, History and Philosophy of Science, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/86e9e38e6963683704881e86f770fa66d4ac39bf

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Citations in the corpus

Tamagawa numbers for motives with (non-commutative) coefficients (2001, result): “The main result of [25] is equivalent to the validity of Conjecture 4(iv) for pairs (h(Spec (L))(r),M(2)) where here G is abelian, M(2) denotes the maximal Z[ 1 2 ]-order in Q[G] and r is any integer.”
On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II. (2006, unknown): “1] and also refines the main result of Huber and Kings in [25] (for more details of the relationship between our approach and that of [25] see [11, Intro.”
The Equivariant Tamagawa Number Conjecture: A survey (2003, methodology): “1 (but only for A a maximal order and l = 2) has been given by Huber and Kings in [45].”
THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES OF MODULAR FORMS (2004, background): “The conjecture is also known for Dirichlet L-functions (including the Riemann zeta function) at any integer [62,4,52,8,35].”
On the equivariant Tamagawa number conjecture for Abelian extensions of a quadratic imaginary field (2006, methodology): “Finally we mention recent work of Huber and Kings [11].”