Kai-Sheng Lei

Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem

We study the arithmetic of the elliptic curves \[ A_m:y^2=x(x-m)(x+4m) \] attached to the $3/5$ congruent number problem. A difference of ternary representation numbers controls the relevant central $L$-values. For $p\equiv11\pmod{40}$ the ordinary Cassels pairing degenerates; we construct an explicit $4$-cover and sho...