Bipartite Euler Systems for certain Galois Representations

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve with ordinary reduction at a prime $p$, and let $K$ be an imaginary quadratic field. The anticyclotomic Iwasawa main conjecture, depending upon the sign of the functional equation of $L(E/K,s)$, predicts the behavior of Selmer group of $E/\mathbb{Q}$ along the anticyclotomic tower of $K$. Some of the crucial ideas of Bertolini and Darmon on this conjecture have been abstracted by Howard into an axiomatic set-up through a notion of Bipartite Euler systems, assuming that $E[p]$ is an irreducible representation of $G_{K}$. We generalize this work by assuming only $(E[p])^{G_K}=0$. We use the results of Howard, Nekov\'a\v{r} and Castella \emph{et al}., along with those of Mazur and Rubin on Kolyvagin systems to show one divisibility of the anticyclotomic main conjecture, for both the signs. The other divisibility can be reduced to proving the nonvanishing of sufficiently many $p$-adic $L$-functions attached to a family of congruent modular forms.