English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The purpose of this paper is to describe a unified method of construction of elliptic curves with given Mordell-Weil rank, and to show it is powerful enough to produce every known example in principle. In view of the fact that there is no general algorithm to give an elliptic curve with high rank, our method might shed a light on this area of active research. The main ingredient in this paper is the twist theory as is developed in [3]. One of our main theorems says that the twists give us every elliptic curve with high Mordell-Weil rank. Once this is established, we naturally come to consider a variety Vn parametrizing a family of elliptic curves with a given rank n, and we show that almost every rational point on this variety gives an elliptic curve with rank ≥ n. Thus our method should provide us with every known example as a rational point on it. We will show that this is indeed the case. In this paper, we parametrize the rational points on Vn in each case of n = 1 to 7. As is mentioned above parametrizing the rational points on Vn is equivalent to getting every elliptic curve of rank at least n (n ≤ 7). Moreover, the variety Vn is expected to be useful for solving other problems. For example, in [5] it is used to construct a family of elliptic curves of rank 2 with given j-invariant. Moreover in [6], it is used to give a family of elliptic curves of rank 6 with a nontrivial rational two-torsion point. The point which is worthy of special mention is that we get very easily the equation of any elliptic curve which corresponds to a rational point on Vn.

A unified method of construction of elliptic curves with high Mordell–Weil rank