# On a non-abelian analogue of a conjecture of Michael Stoll

## TL;DR

This paper proposes a non-abelian generalisation of a conjecture related to Kim's non-abelian Chabauty method, proves the rank 0 quadratic case, and determines the structure of the quadratic Chabauty locus for certain elliptic curves.

## Contribution

It introduces a non-abelian analogue of Stoll's conjecture and proves the rank 0 quadratic case, advancing understanding of non-abelian Chabauty techniques.

## Key findings

- Proved the rank 0 quadratic case of the non-abelian conjecture.
- Determined the structure of the quadratic Chabauty locus for rank 0 elliptic curves.
- Connected the problem to known results about the Manin--Mumford Conjecture.

## Abstract

We formulate a non-abelian generalisation of a conjecture of Stoll, which conjecturally describes the structure of the loci cut out by Kim's method of non-abelian Chabauty. We prove the rank 0 quadratic case of this conjecture, which in particular determines the structure of the quadratic Chabauty locus for once-punctured elliptic curves of rank 0. The proof involves using a variant of the geometric quadratic Chabauty method of Edixhoven and Lido to reduce to an unlikely intersections problem, and ultimately to known results about the relative Manin--Mumford Conjecture.