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Eisenstein degeneration of Euler systems

David Loeffler, Óscar Rivero

Abstract

We discuss the theory of Coleman families interpolating critical-slope Eisenstein series. We apply it to study degeneration phenomena at the level of Euler systems. In particular, this allows us to prove relations between Kato elements, Beilinson--Flach classes and diagonal cycles, and also between Heegner cycles and elliptic units. We expect that this method could be extended to construct new instances of Euler systems.

Eisenstein degeneration of Euler systems

Abstract

We discuss the theory of Coleman families interpolating critical-slope Eisenstein series. We apply it to study degeneration phenomena at the level of Euler systems. In particular, this allows us to prove relations between Kato elements, Beilinson--Flach classes and diagonal cycles, and also between Heegner cycles and elliptic units. We expect that this method could be extended to construct new instances of Euler systems.
Paper Structure (71 sections, 45 theorems, 126 equations)

This paper contains 71 sections, 45 theorems, 126 equations.

Table of Contents

  1. Critical-slope Eisenstein series and their Galois modules
  2. Notations
  3. Galois representations
  4. Classical cohomology
  5. Etale cohomology
  6. De Rham cohomology
  7. Overconvergent cohomology
  8. Compact support
  9. Non-compact supports
  10. Duality and Atkin--Lehner
  11. P-adic families
  12. Pseudocharacters
  13. Families of representations
  14. Comparison of $\mathcal{O}(U)$-lattices
  15. Duality
  16. ...and 56 more sections

Key Result

Theorem A2.1

If $f$ is a $p$-decent Eisenstein series, there are exactly three isomorphism classes of continuous Galois representations $\rho: G_\mathbf{Q} \to \mathop{\mathrm{GL}}\nolimits_2(L)$ which are unramified at primes $\ell \nmid Np$ and satisfy $\operatorname{tr} \rho(\operatorname{Frob}_\ell^{-1}) = a

Theorems & Definitions (107)

  • Definition A1.1
  • Definition A1.2
  • Remark A1.3
  • Theorem A2.1: Soulé
  • Remark A2.2
  • Proposition A3.1
  • Remark A3.2
  • Remark A4.1
  • Proposition A4.2: bellaiche11a
  • Definition A4.3
  • ...and 97 more