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

TL;DR

This paper explores the automorphism groups of certain finite p-groups linked to elliptic curves, providing explicit formulas and interpretations related to torsion points, and generalizing previous work in the area.

Contribution

It introduces explicit formulas for automorphism group orders of p-groups from elliptic curves and explains their variation across primes, extending prior examples.

Findings

Derived formulas for automorphism group orders of p-groups from elliptic curves.

Connected automorphism group sizes to torsion points and flex points over finite fields.

Showed polynomial variation of these orders on Frobenius sets, explaining nonquasipolynomial behavior.

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.