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

Guido Kings, Johannes Sprang

TL;DR

The paper develops a geometric framework for critical values of Hecke $L$-functions over arbitrary totally complex fields by constructing equivariant coherent Eisenstein-Kronecker classes on CM abelian schemes using the completed Poincaré bundle. It then provides explicit computations of these classes in terms of generalized Eisenstein-Kronecker series and currents, connecting them to the de Rham polylogarithm and CM-type decompositions. The results yield integrality statements for regularized values $L(\chi,0)$ after division by explicit periods, and enable a $p$-adic interpolation measure in the ordinary case, generalizing prior CM-field constructions (Damerell, Shimura, Katz) to the broader CM setting. The work also introduces refined classes to manage integrality and develops a robust link between CM types, periods, and special $L$-values, with potential implications for Deligne-type conjectures and CM motives of rank one.

Abstract

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 Poincaré 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 Poincaré 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.

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

TL;DR

The paper develops a geometric framework for critical values of Hecke -functions over arbitrary totally complex fields by constructing equivariant coherent Eisenstein-Kronecker classes on CM abelian schemes using the completed Poincaré bundle. It then provides explicit computations of these classes in terms of generalized Eisenstein-Kronecker series and currents, connecting them to the de Rham polylogarithm and CM-type decompositions. The results yield integrality statements for regularized values after division by explicit periods, and enable a -adic interpolation measure in the ordinary case, generalizing prior CM-field constructions (Damerell, Shimura, Katz) to the broader CM setting. The work also introduces refined classes to manage integrality and develops a robust link between CM types, periods, and special -values, with potential implications for Deligne-type conjectures and CM motives of rank one.

Abstract

We show that for an arbitrary totally complex number field the (regularized) critical -values of algebraic Hecke characters of 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 Poincaré bundle on an abelian scheme . From this we obtain a cohomology class for the automorphism group of a CM abelian scheme 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 Poincaré bundle, we construct a -adic measure interpolating the critical -values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and -adic interpolation in the remaining open cases of critical values of Hecke -functions.

Paper Structure

This paper contains 38 sections, 68 theorems, 614 equations.

Table of Contents

  1. Abelian schemes with CM
  2. CM types
  3. Abelian schemes with CM
  4. Decomposition results and set up
  5. Equivariant coherent Eisenstein-Kronecker classes
  6. Preliminaries on the completed Poincaré bundle
  7. Properties of the completed Poincaré bundle
  8. The equivariant coherent Eisenstein-Kronecker class
  9. The Eisenstein-Kronecker class for CM abelian varieties
  10. A refinement of the Eisenstein-Kronecker class
  11. Proof of Theorem \ref{['thm:coh-with-support-descr']}
  12. Relation to the de Rham polylogarithm
  13. Explicit computation of the equivariant coherent Eisenstein--Kronecker classes
  14. The completed Poincaré bundle as $\mathcal{C}^{\infty}$-bundle
  15. A model to compute equivariant cohomology
  16. ...and 23 more sections

Key Result

Theorem 1

Let $\chi$ be a critical algebraic Hecke character of $L$ (see Section section:cm-types). Let $L(\chi,0)$ be the value of its $L$-function at $0$ and $\mathfrak{c}$ be an integral ideal in $L$ coprime to $\mathfrak{f}$. Then there is a number field $k$ with ring of integers $\mathscr{O}_k$ such that where $\Omega^{\chi}$ is an explicit product of powers of $2\pi i$ and certain periods of abelian v

Theorems & Definitions (187)

  • Theorem : Integrality of critical values, see Theorem \ref{['thm:special-values']}
  • Corollary : Weak Deligne conjecture, see Corollary \ref{['cor:deligne-conjecture']}
  • Theorem : $p$-adic interpolation, see Theorem \ref{['thm_p-adic-interpolation']}
  • Theorem : Eisenstein-Kronecker class, see Section \ref{['subsection:Eisenstein-Kronecker-class']}
  • Theorem 1.1: Decomposition SGA3
  • Definition 1.2
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Definition 1.6
  • ...and 177 more