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An Euler system for the adjoint of a modular form

David Loeffler, Sarah Livia Zerbes

TL;DR

The paper addresses bounding the adjoint Selmer group of a modular form via an Euler system constructed from Asai (base-changed) Beilinson–Flach data on Hilbert modular surfaces. By embedding the adjoint in a decomposed Asai representation over a real quadratic field, it avoids problematic trivial Euler factors and establishes a Greenberg Selmer bound over the cyclotomic $ extbf{Z}_p$-extension, matching the Iwasawa main conjecture up to a power of $p$. The approach yields explicit norm relations, a Perrin-Riou regulator, and an explicit reciprocity law tying the Euler-system to the primitive $p$-adic $L$-function $L_p( ext{ad } f)^{ ext{cy}}$ (and related twists), culminating in a cotorsion conclusion for the Selmer group. The Dirichlet-character obstruction is removed via an input from Byeon, ensuring the final divisibility bound depends only on the adjoint $p$-adic $L$-function and a $p$-power. Overall, the work advances the program of relating adjoint Selmer groups to $p$-adic $L$-functions and supports the main conjecture in this setting.

Abstract

We construct an Euler system for the adjoint Galois representation of a modular form, using motivic cohomology classes arising from Hilbert modular surfaces. We use this Euler system to give an upper bound for the Selmer group of the adjoint representation over the cyclotomic Zp-extension, which agrees with the predictions of the Iwasawa main conjecture up to powers of p.

An Euler system for the adjoint of a modular form

TL;DR

The paper addresses bounding the adjoint Selmer group of a modular form via an Euler system constructed from Asai (base-changed) Beilinson–Flach data on Hilbert modular surfaces. By embedding the adjoint in a decomposed Asai representation over a real quadratic field, it avoids problematic trivial Euler factors and establishes a Greenberg Selmer bound over the cyclotomic -extension, matching the Iwasawa main conjecture up to a power of . The approach yields explicit norm relations, a Perrin-Riou regulator, and an explicit reciprocity law tying the Euler-system to the primitive -adic -function (and related twists), culminating in a cotorsion conclusion for the Selmer group. The Dirichlet-character obstruction is removed via an input from Byeon, ensuring the final divisibility bound depends only on the adjoint -adic -function and a -power. Overall, the work advances the program of relating adjoint Selmer groups to -adic -functions and supports the main conjecture in this setting.

Abstract

We construct an Euler system for the adjoint Galois representation of a modular form, using motivic cohomology classes arising from Hilbert modular surfaces. We use this Euler system to give an upper bound for the Selmer group of the adjoint representation over the cyclotomic Zp-extension, which agrees with the predictions of the Iwasawa main conjecture up to powers of p.
Paper Structure (28 sections, 22 theorems, 31 equations)

This paper contains 28 sections, 22 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Overview of the results
  4. Relation to other work
  5. Statement of the theorem
  6. Galois representations
  7. Symmetric square $L$-functions
  8. Canonical periods and $p$-adic $L$-functions
  9. The Selmer group
  10. Big image
  11. The main theorem
  12. An Euler system for the symmetric square
  13. Asai Galois representations
  14. The Euler system classes
  15. Getting rid of the parasitic norm relation factor
  16. ...and 13 more sections

Key Result

Proposition 2.2.1

The critical values (in the sense of Deligne) for $L(\mathop{\mathrm{Sym}}\nolimits^2 f \times \psi, s)$ are the integers in the range $\{1, \dots, k+1\}$ with $(-1)^s = -\psi(-1)$, and the integers in $\{k+2, \dots, 2k + 2\}$ with $(-1)^s = \psi(-1)$. ∎

Theorems & Definitions (51)

  • Proposition 2.2.1
  • Proposition 2.3.1: Schmidt, Ray--Sujatha--Vatsal
  • proof
  • Conjecture 2.4.1: Greenberg's Iwasawa main conjecture for $T$
  • Remark 2.4.2
  • Theorem A
  • Definition 3.1.2
  • Proposition 3.1.4
  • proof
  • Definition 3.2.2
  • ...and 41 more