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 ε2, 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.
Related
- Iwasawa theory of CM elliptic curves at potentially supersingular primes
- Vanishing of Sha(A/K)[P∞] and its consequences for the anticyclotomic Iwasawa theory of GL2-abelian varieties
- On the Iwasawa invariants of admissible quadratic twists of elliptic curves at ordinary Eisenstein primes
- Density of μ-ordinary Primes for K3 Surfaces
- Kolyvagin's conjecture at non-ordinary primes
BibTeX
@misc{lei2026anticyclotomic,
title = {{Anticyclotomic Iwasawa theory at potentially ordinary primes}},
author = {Antonio Lei and Peter Neamti},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03195},
eprint = {2611.03195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03195},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Anticyclotomic Iwasawa theory at potentially ordinary primes
AU - Antonio Lei
AU - Peter Neamti
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03195
UR - https://arxiv.org/abs/2611.03195
AB - 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.
ER -