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Guido Kings

Publications and source records attributed to Guido Kings.

25 records · Page 2Linked to original sources
arXiv2007

p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure

The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.

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arXiv2006

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

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 Bloch-Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups.

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arXiv2002

Equivariant Bloch-Kato conjecture and non-abelian Iwasawa Main Conjecture

This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how "twisting" in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main Conjecture.

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arXiv2000

The Tamagawa number conjecture for CM elliptic curves

In this paper we prove the Tamagawa number conjecture of Bloch and Kato for CM elliptic curves using a new explicit description of the specialization of the elliptic polylogarithm. The Tamagawa number conjecture describes the special values of the L-function of a CM elliptic curve in terms of the regulator maps of the K-theory of the variety into Deligne and etale cohomology. The regulator map to Deligne cohomology was computed by Deninger with the help of the Eisenstein symbol. For the Tamagawa number conjecture one needs an understanding of the $p$-adic regulator on the subspace of K-theory defined by the Eisenstein symbol. This is accomplished by giving a new explicit computation of the specialization of the elliptic polylogarithm sheaf. It turns out that this sheaf is an inverse limit of $p^r$-torsion points of a certain one-motive. The cohomology classes of the elliptic polylogarithm sheaf can then be described by classes of sections of certain line bundles. These sections are elliptic units and going carefully through the construction one finds an analog of the elliptic Soulé elements. Finally Rubin's ``main conjecture'' of Iwasawa theory is used to compare these elements with etale cohomology.

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arXiv1997

Degeneration of the l-adic Eisenstein symbol and of the elliptic polylog

The main new result is the computation of the degeneration of l-adic Eisenstein classes at the cusps. This is done by relating it to the degeneration of the elliptic polylog. These classes come from K-theory and their Hodge regulator can also be computed (see: Dirichlet motives via modula curves, on the K-theory server). This gives a new proof of a comparison conjecture of Bloch and Kato which was used in the proof of their Tamagawa number conjecture for the Riemann zeta-function. The paper contains appendices on the definition of the classical and elliptic polylog, its degeneration and the comparison to Eisenstein classes.

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