Author

Shi-Song Xu

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Preprint Sep 2026

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...

Kai-Sheng Lei, Shi-Song Xu · 0 citations

Preprint Sep 2026

Positivity and Asymptotics for Chenevier's Orthogonal Polynomials

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 strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel f...

Shi-Song Xu · 0 citations

Preprint Jul 2026

Torsion growth of rational elliptic curves over $\Z_p$-extensions of quadratic fields

Let $E/\Q$ be an elliptic curve and let $\widetilde K_p$ be the compositum of all $\Z_p$-extensions of a quadratic field $K$. We prove that $E(\widetilde K_p)_{\tors}=E(K)_{\tors}$ for $p\geq5$. For $p=3$ and imaginary quadratic $K\neq\Q(\sqrt{-3})$, torsion on each extension is determined by its intersections with the...

Shi-Song Xu, Hao-Yang Yuan · 0 citations