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Eisenstein degeneration of Beilinson--Kato classes and circular units

Javier Polo, Óscar Rivero, Ju-Feng Wu

TL;DR

This work analyzes the degeneration of Beilinson–Kato Euler system elements along Coleman families through the critical Eisenstein point attached to $(\\psi,\\tau)$. It establishes explicit reciprocity laws linking Beilinson–Kato classes to the circular-unit Euler system via a product of $p$-adic $L$-functions and derivatives, under parity and regularity assumptions, and interprets the leading term as an Euler-system incarnation of Bellaïche–Dasgupta factorization. A central result shows that, in the even–even parity case, the projected Beilinson–Kato class ${\\kappa^*(f_{\\beta})}$ equals a constant multiple of $L_p'(f_{\\beta},\\chi)$ and the Kubota–Leopoldt $p$-adic $L$-function of $\\psi$ times the circular-unit $c(\\tau)(r)$, modulo Euler-factor corrections. The paper also develops a deformation framework, discusses cases where the Kato class vanishes or requires leading-term analysis, and sketches extensions to Beilinson–Flach classes and triple-product settings, highlighting the interplay between Euler systems, Iwasawa theory, and $p$-adic $L$-function factorization in Eisenstein contexts.

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 $(ψ,τ)$. 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 Bellaïche and Dasgupta. One of the most significant aspects is that, depending on the parity of $ψ$ and $τ$, 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.

Eisenstein degeneration of Beilinson--Kato classes and circular units

TL;DR

This work analyzes the degeneration of Beilinson–Kato Euler system elements along Coleman families through the critical Eisenstein point attached to . It establishes explicit reciprocity laws linking Beilinson–Kato classes to the circular-unit Euler system via a product of -adic -functions and derivatives, under parity and regularity assumptions, and interprets the leading term as an Euler-system incarnation of Bellaïche–Dasgupta factorization. A central result shows that, in the even–even parity case, the projected Beilinson–Kato class equals a constant multiple of and the Kubota–Leopoldt -adic -function of times the circular-unit , modulo Euler-factor corrections. The paper also develops a deformation framework, discusses cases where the Kato class vanishes or requires leading-term analysis, and sketches extensions to Beilinson–Flach classes and triple-product settings, highlighting the interplay between Euler systems, Iwasawa theory, and -adic -function factorization in Eisenstein contexts.

Abstract

The aim of this note is to explore the Euler system of Beilinson--Kato elements in families passing through the critical -stabilization of an Eisenstein series attached to two Dirichlet characters . 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 -adic -functions vanish. In that regard, our main results may be seen as an Euler system incarnation of the factorization formula of Bellaïche and Dasgupta. One of the most significant aspects is that, depending on the parity of and , 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.
Paper Structure (33 sections, 26 theorems, 135 equations, 1 table)

This paper contains 33 sections, 26 theorems, 135 equations, 1 table.

Table of Contents

  1. Introduction
  2. Set-up
  3. Main results
  4. Relation with other works
  5. Acknowledgements
  6. Galois representations and periods
  7. Eisenstein series and Galois representations
  8. Families of representations
  9. Crystalline periods
  10. Circular units
  11. Beilinson--Kato elements
  12. The Kitagawa--Mazur $p$-adic $L$-function
  13. The adjoint $p$-adic $L$-function
  14. Hida--Rankin vs Kitagawa--Mazur
  15. Kato's Euler system
  16. ...and 18 more sections

Key Result

Theorem 1.3

Suppose that $r$ is even and that the characters $\psi$ and $\tau$ are both even. Then, under Assumption ass-1 and Assumption ass-2, the following equality holds in $H^1(\mathbf{Q}, \mathbf{Q}_p(\tau^{-1})(1+r) \otimes \mathcal{H}_{\Gamma}(-{\mathbf j}))$: where $C$ is a constant which does not depend on $j$.

Theorems & Definitions (88)

  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Soulé
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 78 more