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Anticyclotomic Euler system over biquadratic fields

Kim Tuan Do

TL;DR

This work constructs a new anticyclotomic Euler system for the conjugate-self-dual Galois representation $V_{f,χ}$ attached to a weight-$k$ newform twisted by an anticyclotomic character over an imaginary biquadratic field. The construction begins with weight $(2,2,2)$ diagonal cycles and extends to general weights via Hida families, establishing tame and wild norm relations in the ring-class and anticyclotomic $\mathbf{Z}_p^2$-extensions. The authors derive reciprocity laws tying Euler-system classes to triple-product $p$-adic $L$-functions, and use the JNS framework to obtain Bloch–Kato results in rank $0$, along with divisibility toward the anticyclotomic Iwasawa main conjecture and consequences in rank $1$. These results extend the Heegner-point paradigm to a biquadratic CM setting, providing new evidence for deep conjectures in arithmetic of automorphic representations and Iwasawa theory, with explicit Selmer- and $L$-function connections.

Abstract

We construct a new Euler system (anticyclotomic, in the sense of Jetchev-Nekovar-Skinner) for the Galois representation $V_{f,χ}$ attached to a newform $f$ of weight $k\geq 2$ twisted by an anticyclotomic Hecke character $χ$ defined over an imaginary biquadratic field $K_0$. We then show some arithmetic applications of the constructed Euler system, including results on the Bloch-Kato conjecture and a divisibility towards the Iwasawa-Greenberg main conjecture for $V_{f,χ}$.

Anticyclotomic Euler system over biquadratic fields

TL;DR

This work constructs a new anticyclotomic Euler system for the conjugate-self-dual Galois representation attached to a weight- newform twisted by an anticyclotomic character over an imaginary biquadratic field. The construction begins with weight diagonal cycles and extends to general weights via Hida families, establishing tame and wild norm relations in the ring-class and anticyclotomic -extensions. The authors derive reciprocity laws tying Euler-system classes to triple-product -adic -functions, and use the JNS framework to obtain Bloch–Kato results in rank , along with divisibility toward the anticyclotomic Iwasawa main conjecture and consequences in rank . These results extend the Heegner-point paradigm to a biquadratic CM setting, providing new evidence for deep conjectures in arithmetic of automorphic representations and Iwasawa theory, with explicit Selmer- and -function connections.

Abstract

We construct a new Euler system (anticyclotomic, in the sense of Jetchev-Nekovar-Skinner) for the Galois representation attached to a newform of weight twisted by an anticyclotomic Hecke character defined over an imaginary biquadratic field . We then show some arithmetic applications of the constructed Euler system, including results on the Bloch-Kato conjecture and a divisibility towards the Iwasawa-Greenberg main conjecture for .
Paper Structure (27 sections, 22 theorems, 155 equations)

This paper contains 27 sections, 22 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Relation to previous works
  3. Acknowledgements
  4. Preliminaries
  5. Galois representations associated to newforms
  6. Patched CM Hecke modules
  7. Hecke characters and theta series
  8. Hecke algebras and norm maps
  9. The construction
  10. Digression to primes decomposition and the top left-corner notations
  11. Construction in weight $(2,2,2)$ and tame norm relation
  12. Projection to ring class groups
  13. Construction for general weights and wild norm relations
  14. Selmer groups
  15. Triple product p-adic L-function and Selmer group
  16. ...and 12 more sections

Key Result

Theorem A

There exists a collection of Iwasawa cohomology classes indexed by the ideals $\mu_3\in\mathcal{N}$ with $m=N_{K_3/\mathbf{Q}}(\mu_3)$, where $T_{f,\chi}$ is a certain $G_K$-stable $\mathcal{O}$-lattice inside $V_{f,\chi}$, such that for every prime $\mathop{\mathrm{\lambda}}\nolimits_3\in \mathcal{N}$ of norm $\ell$, with $(\ell,mp)=1$ we have the norm where $P_{{\mathcal{L}}_4}(X)=\det(1-X\cdo

Theorems & Definitions (46)

  • Theorem A: Theorem \ref{['maintheorem2']}
  • Remark
  • Theorem B: Theorem \ref{['thm:BK-def']}
  • Theorem C: Theorem \ref{['thm:BK-def-1']}
  • Proposition A: Proposition 3.2.1 in LLZ-K
  • Theorem B: Corollary 5.2.6 in LLZ-K
  • Proposition A
  • proof
  • Lemma B
  • Proposition C
  • ...and 36 more