On several families of elliptic curves with arbitrary large Selmer groups

TL;DR

This paper calculates Selmer groups for specific families of elliptic curves, demonstrating that these groups can be arbitrarily large, which advances understanding of their structure and size.

Contribution

It provides explicit calculations of Selmer groups for certain elliptic curves, showing they can be made arbitrarily large, a novel result in the field.

Findings

Selmer groups can be arbitrarily large for these elliptic curve families

Explicit descent calculations for these curves

Demonstrates the diversity of Selmer group sizes

Abstract

In this paper, we calculate the $ \phi (\hat{\phi})-$Selmer groups $ S^{(\phi)} (E / \Q) $ and $ S^{(\hat{\varphi})} (E^{\prime} / \Q) $ of elliptic curves $ y^{2} = x (x + \epsilon p D) (x + \epsilon q D) $ via descent theory (see [S, Chapter X]), in particular, we obtain that the Selmer groups of several families of such elliptic curves can be arbitrary large.