Mathematische Annalen https://doi.org/10.1007/s00208-021-02193-8 Mathematische Annalen Hessian matrices, automorphisms of p-groups, and torsion points of elliptic curves Mima Stanojkovski1 · Christopher Voll2 Received: 21 December 2019 / Revised: 13 April 2021 / Accepted: 22 April 2021 © The Author(s) 2021 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. Mathematics Subject Classification 20D15 · 11G20 · 14M12 Contents 1 Introduction and main results ...................................... 2 Groups and Lie algebras from (symmetric) forms ............................ 3 Automorphisms and torsion points of elliptic curves .......................... 4 Degeneracy loci and automorphisms of p-groups ............................ 5 Proofs of the main results and their corollaries ............................. References .................................................. Communicated by Andreas Thom. B Christopher Voll
[email protected] Mima Stanojkovski
[email protected] 1 Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany 2 Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany 123 M. Stanojkovski, C. Voll 1 Introduction and main results In the study of general questions about finite p-groups it is frequently beneficial to focus on groups in natural families.