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.
Highlights
Introduction and main resultsIn the study of general questions about finite p-groups it is frequently beneficial to focus on groups in natural families
In this paper we use this approach to compute the orders of the automorphism groups of groups and Lie algebras GB(F) resp. gB(F) defined in terms of a matrix of linear forms B, where F is a finite field of odd characteristic
In the case that B is a Hessian determinantal representation of an elliptic curve, we give an explicit formula for | Aut(gB(F))|; up to a scalar, this formula gives | Aut(GB(F))|
Summary
In the study of general questions about finite p-groups it is frequently beneficial to focus on groups in natural families. In the case that B is a Hessian determinantal representation of an elliptic curve, we give an explicit formula for | Aut(gB(F))|; up to a scalar, this formula gives | Aut(GB(F))|. We apply this result to a parametrized family of elliptic curves and interpret this formula in terms of arithmetic invariants of the relevant curves; cf Theorem 1.4. Theorem 1.1 Let E be an elliptic curve over Q and let F be a finite field of odd characteristic p over which E has good reduction.
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