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arXiv · 1912.03657

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

Guido Kings·Johannes Sprang
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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 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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Guido Kings, Johannes Sprang. 2019-12-08. Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation. https://arxiv.org/abs/1912.03657

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