Anticyclotomic Iwasawa theory at potentially ordinary primes

Abstract

We study anticyclotomic Iwasawa theory for self-dual twists of $p$-ordinary modular forms of weight two with nebentype $ε^2$, where $ε$ is a non-trivial Dirichlet character of conductor $p$. Under a generalized Heegner hypothesis and suitable residual hypotheses, we prove one divisibility in the anticyclotomic main conjecture in both the definite and indefinite settings. The prime $p$ may split, remain inert, or ramify in the imaginary quadratic field. Our proofs use a compatible bipartite Euler system of Heegner classes and theta elements, together with a comparison with the $p$-adic $L$-functions of Kohen--Pacetti. In the definite case, we prove cotorsion of the Selmer group and the characteristic-ideal divisibility by the $p$-adic $L$-function. In the indefinite case, we prove a rank-one statement and the corresponding divisibility in terms of the Heegner class. We also discuss criteria for equality and their relation to Kolyvagin's conjecture. Specialization gives finite-level Bloch--Kato Selmer bounds with local terms, and applications to Mordell--Weil ranks and $p$-primary Tate--Shafarevich groups for elliptic curves that have potentially ordinary reduction at $p$ satisfying our hypotheses.