# Local torsion primes and the class numbers associated to an elliptic curve over $\mathbb{Q}$

## TL;DR

This paper establishes a lower bound for the class number of certain number fields generated by division points of an elliptic curve over rationals, based on the Mordell-Weil rank and local torsion conditions.

## Contribution

It provides a novel lower bound for class numbers of fields generated by p^n-division points of elliptic curves, under specific local torsion constraints.

## Key findings

- Lower bound for class numbers of $Q(E[p^n])$ fields.
- Relation between Mordell-Weil rank and class number growth.
- Conditions on local torsion points influence class number estimates.

## Abstract

Using the rank of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve $E$ over $\mathbb{Q}$, we give a lower bound of the class number of the number field $\mathbb{Q}(E[p^n])$ generated by $p^n$-division points of $E$ when the curve $E$ does not possess a $p$-adic point of order $p$: $E(\mathbb{Q}_p)[p] =0$.