A Graphical Approach to Computing Selmer Groups of Congruent Number Curves

Sn = {d ∈ M | Cd(Qp) 6= ∅ ∀p|2n, Cd(Q∞) 6= ∅}, S ′ n = {d ∈ M | C ′ d(Qp) 6= ∅ ∀p|2n, C ′ d(Q∞) 6= ∅}, where the equations Cd and C ′ d, in variables (w, t, z) are given by Cd : dw 2 = t + (2n/d)z, C ′ d : dw 2 = t − (n/d)z. We should note that (0, 0, 0) is always a solution to Cd(C ′ d). So, when we write Cd(Qp) 6= ∅ (C ′ d(Qp) 6= ∅), we mean there exists nontrivial solutions. There has been much interest in understanding these groups (see [2, 3, 4, 5] and references there in). In a recent paper of Feng and Xiong ([1]), graph theory is used to describe conditions such that Sn and S ′ n are trivial, which in turn implies that the rank of En is zero. In this paper we use graph theoretic concepts similar to those introduced in [1] to compute Sn and S ′ n.