Jack A. Thorne

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math.NT2025

Lower bounds on heights of odd degree points of hyperelliptic curves

Jef Laga, Jack A. Thorne

We develop a reduction theory for the representation of $SL_{n}$ on pairs of symmetric $n \times n$ matrices. We apply this theory to the pencils of quadrics arising from div…

math.NT2025

Kummers, spinors, and heights

Jef Laga, Jack A. Thorne

Let $f (x) = x^{2 g + 1} + c_{1} x^{2 g} + \dots + c_{2 g + 1} \in k [x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(…

math.NT2025

An LLL algorithm with symmetries

Beth Romano, Jack A. Thorne

We give a generalisation of the Lenstra-Lenstra-LovÃ¡sz (LLL) lattice-reduction algorithm that is valid for an arbitrary (split, semisimple) reductive group $G$ . This can be regard…

math.NT2024

100% of odd hyperelliptic Jacobians have no rational points of small height

Jef Laga, Jack A. Thorne

We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $Q$ . We relate the heights of $Q$ -points of Jacobians of curves in this family…