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Guido Kings

Publications and source records attributed to Guido Kings.

At least 19 recordsLinked to original sources
arXiv2026

$p$-adic $L$-functions for Hecke characters of totally imaginary fields

We construct $p$-adic $L$-functions interpolating critical $L$-values of algebraic Hecke characters for arbitrary unramified primes $p$ and any totally imaginary field. For non-ordinary primes, the only previously known case was that of imaginary quadratic extensions of $\mathbb{Q}$. One of the main ingredients is a new $p$-adic Fourier theory relating generic fibers of $p$-divisible groups to a general class of character varieties. Combining this with equivariant cohomology classes constructed in a previous paper allows us to construct the $p$-adic $L$-function.

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arXiv2025

Algebraicity of critical Hecke $L$-values

In this survey, we review the known results on the algebraicity of critical values of Hecke $L$-functions and explain the new developments in \cite{Kings-Sprang}.

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arXiv2024

Another look at $p$-adic Fourier-theory

In this short note, we show that a natural generalization of the $p$-adic Fourier theory of Schneider and Teitelbaum follows immediately from the classification of $p$-divisible groups over $\cal{O}_{\mathbb{C}_p}$ by Scholze and Weinstein. This paper has been superseded by [arXiv:2603.15446], where the results are extended and generalized.

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arXiv2019

Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation

We show that for an arbitrary totally complex number field $L$ the (regularized) critical $L$-values of algebraic Hecke characters of $L$ divided by certain periods are algebraic integers. This relies on a new construction of an equivariant coherent cohomology class with values in the completion of the Poincar\'e bundle on an abelian scheme $\cal{A}$. From this we obtain a cohomology class for the automorphism group of a CM abelian scheme $\cal{A}$ with values in some canonical bundles, which can be explicitly calculated in terms of Eisenstein-Kronecker series. As a further consequence, using an infinitesimal trivialization of the Poincar\'e bundle, we construct a $p$-adic measure interpolating the critical $L$-values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and $p$-adic interpolation in the remaining open cases of critical values of Hecke $L$-functions.

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arXiv2018

The Maillot-Rössler current and the polylogarithm on abelian schemes

We give a conceptual proof of the fact that the realisation of the degree zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut-Köhler higher analytic torsion form of the Poincaré bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincaré bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.

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arXiv2017

Rankin--Eisenstein classes for modular forms

In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms.

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arXiv2016

On $p$-adic interpolation of motivic Eisenstein classes

In this paper we prove that the motivic Eisenstein classes associated to polylogarithms of commutative group schemes can be $p$-adically interpolated in étale cohomology. This generalizes results for elliptic curves obtained in our former work. Already for one dimensional commutative groups the results proved here are much more flexible as they allow to treat degeneration questions easily.

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arXiv2015

Rankin-Selberg Euler systems and p-adic interpolation

We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.

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arXiv2015

Rankin--Eisenstein classes and explicit reciprocity laws

We construct three-variable $p$-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 $L$-value does not vanish.

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arXiv2014

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

We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for $GL_n(\mathbf{Z})$ and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of $L$-functions of totally real fields and a construction of the associated $p$-adic $L$-function. Using a result of Graf, we also apply this to prove some integrality and $p$-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.

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arXiv2013

Eisenstein classes, elliptic Soulé elements and the $\ell$-adic elliptic polylogarithm

This is a completely rewritten version of the paper formerly entitled "Sheaves of Iwasawa modules, moment maps and the $\ell$-adic elliptic polylogarithm". The proof of the main result is also simplified. In the paper we study systematically the $\ell$-adic realization of the elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads to a description of the elliptic polylogarithm in terms of elliptic units. As an application we prove a precise relation between $\ell$-adic Eisenstein classes and elliptic Soulé elements. This allows to give a new proof of the formula for the residue of the $\ell$-adic Eisenstein classes at the cusps and reproves the formula for the cup-product construction in \cite{Huber-Kings99}. The paper is the elaboration of lectures given at the Pune-Workshop on the proof of the Bloch-Kato conjectures for the Riemann zeta function in 2012.

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arXiv2010

Some complements to the Lazard isomorphism

Lazard showed in his seminal work "Groupes analytiques $p$-adiques" that for rational coefficients continuous group cohomology of $p$-adic Lie-groups is isomorphic to Lie-algebra cohomology. We refine this result in two directions: firstly we extend his isomorphism under certain conditions to integral coefficients and secondly, we show that for algebraic groups, his isomorphism can be realized by differentiating locally analytic cochains.

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arXiv2009

On the equivariant and the non-equivariant main conjecture for imaginary quadratic fields

The Iwasawa main conjecture fields has been an important tool to study the arithmetic of special values of $L$-functions of Hecke characters of imaginary quadratic fields. To obtain the finest possible invariants it is important to know the main conjecture for all prime numbers $p$ and also to have an equivariant version at disposal. In this paper we first prove the main conjecture for imaginary quadratic fields for all prime numbers $p$, improving earlier results by Rubin. From this we deduce the equivariant main conjecture in the case that a certain $μ$-invariant vanishes. For prime numbers $p\nmid 6$ which split in $K$, this is a theorem by a result of Gillard.

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arXiv2009

A cohomological Tamagawa number formula

For smooth linear group schemes over $\bbZ$ we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori the cohomological and the motivic Tamagawa measures coincide, which reproves the Bloch-Kato conjecture for motives associated to tori.

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arXiv2007

A note on polylogarithms on curves and abelian schemes

In this note we investigate the connection between polylogarithms on curves and abelian schemes. The main result shows that the polylogarithm on the abelian scheme can be obtained as the push-forward of the polylogarithm on a suitable sub-curve. If the abelian scheme is the Jacobian of a smooth projective curve, this push-forward can also be written as a cup-product with the fundamental class of the curve.

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