Hessian matrices, automorphisms of $p$-groups, and torsion points of elliptic curves

Abstract

We describe the automorphism groups of finite $p$-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of $j$-invariant $1728$ given in Weierstrass form. We interpret these orders in terms of the numbers of $3$-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arising in these examples are polynomial on Frobenius sets and vary with the primes in a nonquasipolynomial manner.

27 pages, minor revisions, including referee's suggestions. To appear in Mathematische Annalen

References in corpus (4)

• A fast isomorphism test for groups of genus 2
• Skolem-Noether for nilpotent products
• The average size of the kernel of a matrix and orbits of linear groups, II: duality
• Simple determinantal representations of up to quintic bivariate polynomials