Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms

2012/12/13 by Olivier Fouquet · 25 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number field #Algebraic structures and combinatorial models #Conjecture #Euler system #Galois group #Galois module #Geometry #Iwasawa theory #Mathematical analysis #Mathematics #Pure mathematics #Quadratic equation #Quadratic field

paper · pdf · doi:10.1112/s0010437x12000619

published in Compositio Mathematica 149(3), 356-416 (Cambridge University Press)

openalex publication_date 2012/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Abstract Let π ( f ) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρ f . Let K be a totally complex quadratic extension of F embedding in B . Using families of CM points on towers of Shimura curves attached to B and K , we construct an Euler system for ρ f . We prove that it extends to p -adic families of Galois representations coming from Hida theory and dihedral ℤ d p -extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.

Citations

• On ordinary ?-adic representations associated to modular forms
• Good Reduction of Abelian Varieties

Cited by

• On anticyclotomic Euler and Kolyvagin systems
• Quaternionic big Heegner points over totally real fields
• Quaternionic families of Heegner points and p-adic L-functions
• Quaternion algebras, Heegner points and the arithmetic of Hida families
• Iwasawa Main Conjecture for Heegner Points: Supersingular Case
• A refined Beilinson-Bloch conjecture for motives of modular forms
• Quaternionic Kolyvagin systems and Iwasawa theory for Hida families
• The Iwasawa Main Conjectures for \rm GL₂ and derivatives of p-adic L-functions
• Deformations of Kolyvagin systems
• Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes
• The -adic Gross–Zagier formula on Shimura curves
• The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in the Hecke algebra

Related

• Quaternionic Kolyvagin systems and Iwasawa theory for Hida families
• Trito-non-ordinary Iwasawa theory of diagonal cycles
• On certain cohomology groups attached to \mathfrakp^∞-towers of quaternionic Hilbert modular varieties
• Note sur les diviseurs élémentaires du régulateur d'Iwasawa
• On the Iwasawa Main Conjecture for generalized Heegner classes in a quaternionic setting