# Quadratic Chabauty for modular curves and modular forms of rank one

## TL;DR

This paper refines the quadratic Chabauty method to ensure finiteness of rational points on certain modular curves by using conditions on Jacobian quotients and applies it to prove finiteness for specific modular curves of genus at least 2.

## Contribution

It introduces new conditions involving Jacobian quotients and Chow-Heegner points for the quadratic Chabauty method and applies these to prove finiteness results for modular curves of rank at least 2.

## Key findings

- Finiteness of rational points on certain modular curves established.
- Refined conditions for quadratic Chabauty method to produce finite sets.
- Existence of Jacobian quotients with Mordell-Weil rank equal to dimension proven.

## Abstract

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin-Logachev type result.