On the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ - Dx

Abstract

We study the variation of the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integer D is constrained to lie in a family for which the relative Tamagawa number of the isogeny $\phi$ is fixed.