Quadratic Chabauty and $p$-adic Gross-Zagier

TL;DR

This paper develops a novel $p$-adic analytic method using quadratic Chabauty and Gross-Zagier formulas to determine rational points on certain modular curves without prior knowledge of rational points.

Contribution

It introduces an algorithm to compute special values of anticyclotomic $p$-adic $L$-functions, enabling quadratic Chabauty to find rational points without initial point data.

Findings

Successfully computes rational points on modular curves.

Eliminates the need for known rational points in quadratic Chabauty.

Provides a new computational approach for anticyclotomic $p$-adic $L$-functions.

Abstract

Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a simple factor of $J_0(N)^{new}$ over $\mathbb{Q}$. Let $f$ be the newform of level $N$ and weight 2 associated with $J_X$; assume $f$ has analytic rank 1. We give analytic methods for determining the rational points of $X$ using quadratic Chabauty by computing two $p$-adic Gross-Zagier formulas for $f$. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic $p$-adic $L$-function of $f$ constructed by Bertolini, Darmon, and Prasanna, which lies outside of the range of interpolation.