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An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation

Raúl Alonso, Francesc Castella, Óscar Rivero

Abstract

We construct an anticyclotomic Euler system for the Asai Galois representation associated to $p$-ordinary Hilbert modular forms over real quadratic fields. We also show that our Euler system classes vary in $p$-adic Hida families. The construction is based on the study of certain Hirzebruch--Zagier cycles obtained from modular curves of varying level diagonally emdedded into the product with a Hilbert modular surface. By Kolyvagin's methods, in the form developed by Jetchev--Nekovář--Skinner in the anticyclotomic setting, the construction yields new applications to the Bloch--Kato conjecture and the Iwasawa Main Conjecture.

An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation

Abstract

We construct an anticyclotomic Euler system for the Asai Galois representation associated to -ordinary Hilbert modular forms over real quadratic fields. We also show that our Euler system classes vary in -adic Hida families. The construction is based on the study of certain Hirzebruch--Zagier cycles obtained from modular curves of varying level diagonally emdedded into the product with a Hilbert modular surface. By Kolyvagin's methods, in the form developed by Jetchev--Nekovář--Skinner in the anticyclotomic setting, the construction yields new applications to the Bloch--Kato conjecture and the Iwasawa Main Conjecture.
Paper Structure (31 sections, 19 theorems, 197 equations)

This paper contains 31 sections, 19 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. The Euler system
  4. Applications
  5. Further directions
  6. Related works
  7. Acknowledgements
  8. Review of Hilbert cuspforms
  9. General notations
  10. Adelic Hilbert cuspforms
  11. Hecke Theory
  12. Galois representations
  13. Asai representations
  14. Hida families
  15. Big Galois representations
  16. ...and 16 more sections

Key Result

Theorem A

Suppose $p$ splits in $\cal{K}$ and $p\nmid h_{\cal{K}}$, and that $g$ is ordinary at $p$. There exists a collection of classes such that whenever $n, nq\in\mathcal{S}$ with $q$ a prime, we have where $\mathfrak{q}$ is a prime of $\mathcal{K}$ above $q$.

Theorems & Definitions (46)

  • Theorem A: Theorem \ref{['thm:ES']}
  • Theorem B: Theorem \ref{['thm:BK']}
  • Theorem C: Theorem \ref{['thm:IMC']}
  • Definition 2.1
  • Remark 1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 36 more