Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Iwasawa theory for quadratic Hilbert modular forms

David Loeffler, Sarah Livia Zerbes

TL;DR

The paper proves a Kato-style divisibility for the Iwasawa main conjecture of quadratic Hilbert modular forms over the p-cyclotomic tower, and, by contrasting with Wan's opposite divisibility, establishes the full Main Conjecture in this setting. It constructs Euler systems via twisted Yoshida lifts to GSp_4, employs eigenvariety deformation to treat parallel-weight points, and develops a leading-term framework to relate Euler-system data to p-adic L-functions and regulator maps. The work yields new instances of the Bloch–Kato conjecture and equivariant BSD in analytic rank 0, with additional by-products for Rankin–Selberg convolutions and p-adic L-functions in families. These results extend the arithmetic reach of Iwasawa theory, providing concrete divisibility and finiteness statements for Selmer groups and new non-vanishing results for twists of L-values across families of automorphic forms.

Abstract

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under certain technical hypotheses. By comparing this result with the opposite divisibility due to Wan, we obtain the full Main Conjecture over the cyclotomic Zp-extension. As a consequence, we prove new cases of the Bloch--Kato conjecture for quadratic Hilbert modular forms, and of the equivariant Birch--Swinnerton-Dyer conjecture in analytic rank 0 for elliptic curves over real quadratic fields twisted by Dirichlet characters. As a "by-product" of the theory developed here, we also present new results on Iwasawa theory for Rankin--Selberg convolutions of modular forms, relaxing hypotheses of $p$-distinction or $p$-regularity assumed in previous works. This gives new cases of the equivariant BSD conjecture for elliptic curves over $\mathbf{Q}$ twisted by 2-dimensional odd Artin representations, giving finiteness of the $p$-part of the Tate--Shafarevich group for all but finitely many ordinary primes.

Iwasawa theory for quadratic Hilbert modular forms

TL;DR

The paper proves a Kato-style divisibility for the Iwasawa main conjecture of quadratic Hilbert modular forms over the p-cyclotomic tower, and, by contrasting with Wan's opposite divisibility, establishes the full Main Conjecture in this setting. It constructs Euler systems via twisted Yoshida lifts to GSp_4, employs eigenvariety deformation to treat parallel-weight points, and develops a leading-term framework to relate Euler-system data to p-adic L-functions and regulator maps. The work yields new instances of the Bloch–Kato conjecture and equivariant BSD in analytic rank 0, with additional by-products for Rankin–Selberg convolutions and p-adic L-functions in families. These results extend the arithmetic reach of Iwasawa theory, providing concrete divisibility and finiteness statements for Selmer groups and new non-vanishing results for twists of L-values across families of automorphic forms.

Abstract

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under certain technical hypotheses. By comparing this result with the opposite divisibility due to Wan, we obtain the full Main Conjecture over the cyclotomic Zp-extension. As a consequence, we prove new cases of the Bloch--Kato conjecture for quadratic Hilbert modular forms, and of the equivariant Birch--Swinnerton-Dyer conjecture in analytic rank 0 for elliptic curves over real quadratic fields twisted by Dirichlet characters. As a "by-product" of the theory developed here, we also present new results on Iwasawa theory for Rankin--Selberg convolutions of modular forms, relaxing hypotheses of -distinction or -regularity assumed in previous works. This gives new cases of the equivariant BSD conjecture for elliptic curves over twisted by 2-dimensional odd Artin representations, giving finiteness of the -part of the Tate--Shafarevich group for all but finitely many ordinary primes.

Paper Structure

This paper contains 87 sections, 68 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Our results
  4. Outline of the paper
  5. Preliminaries I: Hilbert modular forms
  6. Weights
  7. Automorphic representations and L-functions
  8. Periods and rationality of L-values
  9. Betti cohomology
  10. Complex periods
  11. Gauss sums
  12. The $L$-function as a linear map
  13. Non-vanishing of L-values
  14. Galois representations
  15. Big Galois image
  16. ...and 72 more sections

Key Result

Theorem A

Assume that: Then the characteristic ideal of $\widetilde{H}^2_{\mathrm{Iw}}(K_\infty, \rho_{\pi, v}^*)$ divides $L_p(\pi)$ in $\Lambda_{\mathcal{O}}(\Gamma)$. In particular, if $L_p(\pi)$ is not a zero-divisor, then $\widetilde{H}^2_{\mathrm{Iw}}(K_\infty, \rho_{\pi, v}^*)$ is torsion, and $\widetilde{H}^i_{\ma

Theorems & Definitions (186)

  • Conjecture 1.1.1: Bloch--Kato conjecture
  • Conjecture 1.1.2: Iwasawa main conjecture
  • Theorem A: \ref{['thm:withzeta']}
  • Theorem B: \ref{['thm:mainconj']}
  • Theorem C: \ref{['thm:BK']}
  • Theorem D: \ref{['thm:BSD']}
  • Definition 2.1.1
  • Remark 2.1.2
  • Definition 2.2.1
  • Remark 2.2.2
  • ...and 176 more