Bipartite Euler systems

TL;DR

This paper explores the Iwasawa main conjecture for elliptic curves over imaginary quadratic fields, showing it can be reduced to proving nonvanishing of certain p-adic L-functions, regardless of the functional equation sign.

Contribution

It combines ideas from Bertolini-Darmon and Mazur-Rubin to unify the approach to the main conjecture across different functional equation signs.

Findings

Main conjecture reduces to nonvanishing of p-adic L-functions

Unifies cases for different signs of the functional equation

Provides a new approach linking modular forms and Iwasawa theory

Abstract

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different forms depending on the sign of the functional equation of L(E/K,s). In the present work we combine ideas of Bertolini and Darmon with those of Mazur and Rubin to shown that the main conjecture, regardless of the sign of the functional equation, can be reduced to proving the nonvanishing of sufficiently many p-adic L-functions attached to a family of congruent modular forms.