Conjecture A and $\mu$-invariant for Selmer groups of supersingular elliptic curves

TL;DR

This survey explores algebraic Iwasawa theory of Selmer groups for supersingular elliptic curves, focusing on the $mbda$-invariant, Conjecture A, and analogies with ordinary reduction cases.

Contribution

It summarizes recent algebraic results on Selmer groups over cyclotomic and mbda-extensions, emphasizing the $mbda$-invariant and conjectural vanishing results.

Findings

Recent results suggest vanishing of the mbda-invariant under Conjecture A.

Analogies between classical and signed Selmer groups highlight structural similarities.

Properties of signed Selmer groups mirror those of classical Selmer groups in ordinary cases.

Abstract

Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.