On the class numbers of the fields of the p^n-torsion points of certain elliptic curves over Q

TL;DR

This paper investigates the growth of class numbers in fields generated by p^n-torsion points of elliptic curves over Q, providing lower bounds related to the Mordell-Weil rank and an explicit example.

Contribution

It establishes lower bounds for the p-Sylow subgroup of class groups of fields from elliptic curve torsion points, linking class number growth to Mordell-Weil rank and providing explicit divisibility examples.

Findings

Lower bounds for class group p-Sylow groups in torsion fields

Connection between Mordell-Weil rank and class number growth

Explicit example with divisibility by p^{2n} for p=5077

Abstract

Let E be an elliptic curve over Q with prime conductor p. For each non-negative integer n we put K_n:=Q(E[p^n]). The aim of this paper is to estimate the order of the p-Sylow group of the ideal class group of K_n. We give a lower bounds in terms of the Mordell-Weil rank of $E(\Q)$. As an application of our result, we give an example such that p^{2n} divides the class number of the field $K_n$ in the case of $p=5077$ for each positive integer n.