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Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4)

Giada Grossi, David Loeffler, Sarah Livia Zerbes

TL;DR

This work proves the Bloch–Kato conjecture for critical values of Asai L-functions attached to p-ordinary Hilbert modular forms over real quadratic fields (with p split) and establishes one inclusion in the cyclotomic Iwasawa Main Conjecture for these L-functions. Central to the approach is the Asai–Flach Euler system, the construction of a three-variable p-adic Asai L-function via higher Hida theory, and a regulator formula linking Bloch–Kato logarithms to p-adic L-values; these are interpolated in Hida families through a p-adic Eichler–Shimura comparison with meromorphic interpolation. The paper develops a robust framework of coherent and fp-cohomology, plus Poznań spectral sequences, to express regulator pairings explicitly in terms of p-adic L-values and to prove an explicit reciprocity law in families. The results yield leading-term arguments that connect Euler-system data with L-values, enabling proofs of the Bloch–Kato conjecture in the Asai setting, and establishing an Iwasawa-theoretic inclusion that advances our understanding of the arithmetic of GO(4)–type motives. Overall, the work provides a comprehensive, family-wide bridge between Euler systems, p-adic L-functions, and Selmer groups for Asai representations arising from Hilbert modular forms.

Abstract

We prove the Bloch-Kato conjecture for critical values of Asai L-functions of p-ordinary Hilbert modular forms over quadratic fields (with p split); and one inclusion in the Iwasawa main conjecture for these L-functions (up to a power of p). Along the way, we also prove a version of the p-adic Eichler-Shimura comparison isomorphism for Hida families of Hilbert modular forms.

Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4)

TL;DR

This work proves the Bloch–Kato conjecture for critical values of Asai L-functions attached to p-ordinary Hilbert modular forms over real quadratic fields (with p split) and establishes one inclusion in the cyclotomic Iwasawa Main Conjecture for these L-functions. Central to the approach is the Asai–Flach Euler system, the construction of a three-variable p-adic Asai L-function via higher Hida theory, and a regulator formula linking Bloch–Kato logarithms to p-adic L-values; these are interpolated in Hida families through a p-adic Eichler–Shimura comparison with meromorphic interpolation. The paper develops a robust framework of coherent and fp-cohomology, plus Poznań spectral sequences, to express regulator pairings explicitly in terms of p-adic L-values and to prove an explicit reciprocity law in families. The results yield leading-term arguments that connect Euler-system data with L-values, enabling proofs of the Bloch–Kato conjecture in the Asai setting, and establishing an Iwasawa-theoretic inclusion that advances our understanding of the arithmetic of GO(4)–type motives. Overall, the work provides a comprehensive, family-wide bridge between Euler systems, p-adic L-functions, and Selmer groups for Asai representations arising from Hilbert modular forms.

Abstract

We prove the Bloch-Kato conjecture for critical values of Asai L-functions of p-ordinary Hilbert modular forms over quadratic fields (with p split); and one inclusion in the Iwasawa main conjecture for these L-functions (up to a power of p). Along the way, we also prove a version of the p-adic Eichler-Shimura comparison isomorphism for Hida families of Hilbert modular forms.
Paper Structure (59 sections, 59 theorems, 138 equations, 1 figure)

This paper contains 59 sections, 59 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Conventions
  5. Algebraic groups
  6. Algebraic representations and the Clebsch--Gordan map
  7. Setup: Hilbert modular forms
  8. Notations
  9. Automorphic representations of $G$
  10. The Asai $L$-function
  11. Hecke eigensystems
  12. Eigensystems in coherent cohomology
  13. Classes in coherent cohomology
  14. Coherent cohomology of the 1-ordinary locus
  15. Comparison with cohomology at $\mathop{\mathrm{Iw}}\nolimits(\mathfrak{p}_1)$-level
  16. ...and 44 more sections

Key Result

Theorem A

Let $\Pi$ be an automorphic representation of $\mathop{\mathrm{Res}}\nolimits_{F / \mathbb{Q}} \mathop{\mathrm{GL}}\nolimits_2$ of weight $(k_1 + 2, k_2 + 2)$, for some $k_1 > k_2 \geqslant 0$ with $k_1 = k_2 \bmod 2$, and level $\mathfrak{N}$, and $p > 2$ a prime. Suppose that Then:

Figures (1)

  • Figure 1: Relations between the coherent classes at spherical and $\mathfrak{p}_1$-Iwahori level (all cohomology groups with coefficients in $\omega_G^{(-k_1, k_2 + 2)}(-D_G)$)

Theorems & Definitions (136)

  • Theorem A
  • Theorem B: \ref{['thm:regulator']}
  • Theorem C: \ref{['thm:padicL']}
  • Theorem D: \ref{['thm:ERL']}
  • Definition 3.2.1
  • Remark 3.2.2
  • Definition 3.3.1
  • Definition 3.4.1
  • Definition 3.4.5
  • Remark 3.4.6
  • ...and 126 more