Guido Kings

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

This is the completely revised version of math.AG/0101071.

Rankin–Eisenstein classes and explicit reciprocity laws

We construct three-variable

Dirichlet motives via modular curves

p

A p-adic analogue of the Borel regulator and the Bloch-Kato exponential map

-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated

On the equivariant main conjecture for imaginary quadratic fields

L

Higher Analytic Torsion, Polylogarithms and Norm Compatible Elements on Abelian Schemes

-value does not vanish.

A cohomological Tamagawa number formula

Abstract Generalizing ideas of Anderson, Harder has proposed a construction of extensions of Tate-motives (more precisely of Hodge structures and Galois modules, respectively) in terms of modular curves. The aim of this paper is to construct directly those elements of motivic cohomology of Spec Q (μN) (i.e. in K∗(Spec Q (μN))) which induce these extensions in absolute Hodge cohomology and continuous Galois cohomology. We give two such constructions and prove that they are equivalent. The key ingredient is Beilinsons Eisenstein symbol in motivic cohomology of powers of the universal elliptic curve over the modular curve. We also compute explicitly the Harder-Anderson element in absolute Hodge cohomology. It is given in terms of Dirichlet-L-functions. As a corollary, we get a new proof of Beilinsons conjecture for Dirichlet-L-functions. A second paper [HuK] treats the explicit computation in the l-adic case.

Topological polylogarithms and p-adic interpolation of L-values of totally real fields

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the BlochKato exponential and the Soule regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups.

Polylogarithm for families of commutative group schemes

Abstract In this paper we first prove the main conjecture for imaginary quadratic fields for all prime numbers p, improving slightly earlier results by Rubin. From this we deduce the equivariant main conjecture in the case that a certain μ-invariant vanishes. For prime numbers p ∤ 6 which split in K, we can prove the equivariant main conjecture using a theorem by Gillard.

The Tamagawa number conjecture for CM elliptic curves

We give a simple axiomatic description of the degree 0 part of the polylogarithm on abelian schemes and show that its realisation in analytic Deligne cohomology can be described in terms of the Bismut–Kohler higher analytic torsion form of the Poincare bundle.