On the Rank of Elliptic Curves in Elementary Cubic Extensions

Rintaro Kozuma

doi:10.1155/2015/501629

Abstract

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

Highlights

Let E be an elliptic curve defined over a number field F
The present paper is motivated by the following general problem: To understand the behavior of ranks of elliptic curves in towers of finite extensions over F
In the context of Iwasawa theory, Kurcanov [2] showed that if an elliptic curve E over Q without complex multiplication has good reduction at a prime number p > 3 satisfying a mild condition E has infinite rank in a certain Zp-extension, and Harris [3] showed that if E is a modular elliptic curve over F having good ordinary reduction at p and a specific F-rational point arising from the modular curve there is a p-adic Lie extension of F in which the rank of E grows infinitely

Summary

Introduction

Let E be an elliptic curve defined over a number field F It is well-known that the Mordell-Weil group E(F) of F-rational points on E forms a finitely generated abelian group, and its rank rank E(F) = dimQE(F)⊗ZQ is of great interest in arithmetic geometry. The present paper is motivated by the following general problem: To understand the behavior of ranks of elliptic curves in towers of finite extensions over F. This problem dates back at least to [1], which first introduced an Iwasawa theory for elliptic curves. The question of finding a small elementary cubic extension with rank ≥ l seems to be of independent interest

Main Results

A Method for Constructing an Elementary Cubic Extension

Conflict of Interests