Representation Defects and Cassels Pairings for Congruent Number Curves

Qin expressed central values of the congruent number curve $E_n:y^2=x^3-n^{2}x$ through differences of ternary quadratic form representation numbers, while Zhang gave explicit formulas for the Cassels pairing on the pure $2$-Selmer group. We prove that the parity of Qin’s normalized representation defect is the Pfaffian of the Cassels pairing. For $n=pq$ with $s_2=4$, we compute the $4\times 4$ Cassels matrix and obtain an invariant $\mathrm{\Pi }(p,q)$ such that $r_a(pq)\equiv h(-pq)+16\mathrm{\Pi }(p,q)(\mathrm{m}\mathrm{o}\mathrm{d}32)$. We also prove an even analogue and a higher dimensional extension.