What question did this study set out to answer?

The research aims to explore the properties of the dual Selmer group of elliptic curves over specific extensions, particularly focusing on the implications of the Heegner hypothesis.

March 1, 2026Open Access

Topics in anticyclotomic Iwasawa theory

Key Points

The research aims to explore the properties of the dual Selmer group of elliptic curves over specific extensions, particularly focusing on the implications of the Heegner hypothesis.
Examined anticyclotomic extensions of imaginary quadratic fields related to elliptic curves.
Used algebraic techniques to establish the non-existence of finite submodules in specific cases.
Extended existing results on the μ-invariant and λ-invariants using simpler methods.
Conducted an analysis of BDP p-adic L-functions in relation to congruences of modular forms.
Demonstrated the non-existence of non-zero finite submodules under certain conditions.
Established the vanishing of the μ-invariant for elliptic curves related to specific Heegner points.
Observed variations in λ-invariants for p-residually isomorphic elliptic curves.
Investigated the behavior of p-adic L-functions for congruent modular forms, contributing to the analytic side of the study.

Abstract

Let p ≥ 5 be a prime and E/ℚ be an elliptic curve of conductor N that is ordinary at p. Let K/ℚ be an imaginary quadratic field. This thesis is concerned with the dual Selmer group of E over the anticyclotomic extension of K, especially when p is split in K and K satisfies the Heegner hypothesis for E: every prime ℓ dividing N is split in K/ℚ. A fundamental result in this setting is the non-existence of non-zero finite submodules. Using purely algebraic methods, we extend this result to new cases under verifiable hypotheses concerning the Heegner point of E over K. Under similar hypotheses, we establish the vanishing of the μ-invariant using simpler techniques than the literature. As an application, we study the variation of λ-invariants for p-residually isomorphic elliptic curves. This type of congruence question is also explored on the analytic side. Namely, we look at the Bertolini-Darmon-Prasanna (BDP) p-adic L-functions for p-congruent modular forms.

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