# Independence of CM points in Elliptic Curves

## TL;DR

This paper establishes a comprehensive result characterizing linear dependencies among special points on elliptic curves, unifying and enhancing previous findings in the context of modular and Shimura curves.

## Contribution

It provides a unified framework for understanding linear relations among special points on elliptic curves, improving upon prior work by Rosen-Silverman, Kühne, and Buium-Poonen.

## Key findings

- Characterizes all linear dependencies among special points for each n
- Unifies previous results in modular and Shimura curve contexts
- Improves the understanding of special point relations in elliptic curves

## Abstract

We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects previous work of Rosen-Silverman--K\"uhne and Buium-Poonen.