# A result on the analytic $\mu$-invariant of elliptic curves

## TL;DR

This paper extends a known result relating the analytic d-invariants of elliptic curves, showing that isomorphic Galois modules of their torsion subgroups imply a specific relation between their d-invariants.

## Contribution

It generalizes Greenberg and Vatsal's result to broader conditions, linking torsion subgroup isomorphisms to d-invariant relations.

## Key findings

- Established a generalized relation between d-invariants of elliptic curves.
- Connected Galois module isomorphisms of torsion subgroups to d-invariant equality.
- Extended previous results to more general cases.

## Abstract

We generalise a result by Greenberg and Vatsal on the relation between the analytic $\mu$-invariants of two elliptic curves whose $p^i$-torsion subgroups are isomorphic as Galois modules, for suitable $i$.