5 papers
Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem
Kaisheng Lei, Shisong Xu
We study the arithmetic of the elliptic curves \[ A_m:y^2=x(x-m)(x+4m) \] attached to the congruent number problem. A difference of ternary representation numbers controls th…
Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture
Shisong Xu
We construct, over every formally real field, a singular pair of quadratic forms in dimension seven whose adjoint closure equals its pencil and consists entirely of hyperbolic form…
Positivity and Asymptotics for Chenevier's Orthogonal Polynomials
Shisong Xu
We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite--Minkowski theorem. The proof establishes stri…
Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
Shisong Xu
Using BSD results for CM elliptic curves, we relate Qin's quadratic form representation defects to the Cassels pairing on the pure -Selmer group of a congruent number elliptic c…
Torsion growth of rational elliptic curves over -extensions of quadratic fields
Shisong Xu, Haoyang Yuan
Let $E/\Q$ be an elliptic curve and let be the compositum of all -extensions of a quadratic field . We prove that $E(\widetilde K_p)_{\tors}=E(K)_{\tors}$…