On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms

TL;DR

This paper constructs a bipartite Euler system for Hilbert modular eigenforms, advancing the understanding of Iwasawa main conjectures by establishing new divisibility results and linking them to p-adic L-functions.

Contribution

It generalizes previous constructions to Hilbert modular forms over totally real fields and applies these to main conjectures in Iwasawa theory.

Findings

Establishes one divisibility of the dihedral main conjecture in many cases.

Reduces the other divisibility to a nonvanishing criterion for p-adic L-functions.

Applies techniques to cyclotomic main conjectures over CM fields.

Abstract

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.