Eisenstein degeneration of Beilinson--Kato classes and circular units

TL;DR

This paper investigates the relationship between Beilinson--Kato Euler systems and circular units in the context of Eisenstein series, revealing new factorization formulas and phenomena depending on character parity.

Contribution

It establishes an explicit connection between Beilinson--Kato classes and circular units, extending factorization formulas in the setting of Eisenstein series and Euler systems.

Findings

Connection between Beilinson--Kato classes and circular units

Factorization formulas for p-adic L-functions with vanishing components

Parity-dependent phenomena in Euler system behavior

Abstract

The aim of this note is to explore the Euler system of Beilinson--Kato elements in families passing through the critical $p$-stabilization of an Eisenstein series attached to two Dirichlet characters $(\psi,\tau)$. In this context, we establish an explicit connection with the system of circular units, utilizing suitable factorization formulas in a situation where several of the $p$-adic $L$-functions vanish. In that regard, our main results may be seen as an Euler system incarnation of the factorization formula of Bella\"iche and Dasgupta. One of the most significant aspects is that, depending on the parity of $\psi$ and $\tau$, different phenomena arise; while some can be addressed with our methods, others pose new questions. Finally, we discuss analogous results in the framework of Beilinson--Flach classes.