Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields

Abstract

Using BSD results for CM elliptic curves, we relate Qin's quadratic form representation defects to the Cassels pairing on the pure $2$-Selmer group of a congruent number elliptic curve. When the pure $2$-Selmer dimension is even, we prove that the normalized representation defect modulo $2$ is the Pfaffian of the Cassels pairing matrix; the dimension of its radical yields sharper $2$-adic divisibility and information on the $2$-primary Shafarevich--Tate group. For products of primes congruent to $1$ modulo $8$ that are pairwise quadratic residues, we give explicit Cassels pairing matrices for both $E_n$ and $E_{2n}$ and express their entries in terms of quartic and Rédei symbols. For each fixed prime $p$, we construct a governing field of degree $256$ and determine the exact joint distribution of the two Pfaffians by Chebotarev's theorem. In particular, there is a set of primes $q$ of natural density $5/128$ for which both $E_{17q}$ and $E_{34q}$ have rank zero and $2$-primary Shafarevich--Tate group isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^4$.