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

Abstract: I report on joint work with Johannes Sprang. Let $K$ be a CM field and $L/K$ be an extension of degree $n$ and $\chi$ be an algebraic critical Hecke character of $L$. Then we show that the $L$-value $L(\chi, 0)$ divided by carefully normalized Shimura-Katz periods is integral and construct a $p$-adic $L$-function for $\chi$. This generalizes results by Damerell, Shimura and Katz for CM fields ($L = K$) and settles all open cases of algebraicity for critical Hecke $L$-values.

Our method relies on a detailed analysis of new equivariant motivic Eisenstein classes and especially on the study of their de Rham realizations and is completely different from the classical approach by Shimura and Katz. The de Rham realization of these Eisenstein classes can be explicitly described in terms of Eisenstein-Kronecker series and the equivariant setting is crucial to connect them with the $L$-function of $\chi$. An integral refinement of this construction leads directly to a geometric construction of a $p$-adic measure without any need to check congruences for the Eisenstein series.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar