On the Joint Distribution Of Selφ(E/Q) and Selφ(E ′/Q) in Quadratic Twist Families

Strikingly, this is not true when E has a single rational point of order two. In this case E has a degree two isogeny φ : E → E ′ and an associated Selmer group Selφ(E/Q). Work of Xiong shows that if E does not have a cyclic 4-isogeny de ned over Q(E[2]), then the distribution of the ranks of Selφ(E /Q) as d varies among the squarefree integers less than X tends to the distribution Max{0,N (0, 1 2 log logX)} as X → ∞, where N (μ, σ) is the normal distribution with mean μ and variance σ [Xio13]. In this case, Selφ(E/Q) maps 2− to− 1 into Sel2(E/Q), showing that for any xed r, at least half of the quadratic twists of E have 2-Selmer rank greater than r.