On several families of elliptic curves with arbitrary large Selmer groups

Abstract

In this paper, we calculate the $\varphi (\hat{\varphi })-$Selmer groups $ S^{(ϕ)} (E / \Q) $ and $ S^{(\hatφ)} (E^{\prime} / \Q) $ of elliptic curves $y^2=x(x+\epsilon pD)(x+\epsilon qD)$ 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.

22 pages