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

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.


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. Often, this affords an additional geometric point of view on the original group-theoretic questions. When considering, for instance, the members of a family (G(F p )) p prime of groups of F p -rational points of a unipotent group scheme G, it is of interest to understand the interplay between properties of the abstract groups G(F p ) with the structure of the group scheme G. Specifically, geometric insights into the automorphism group Aut(G) of G translate into uniform statements about the automorphism groups Aut(G(F p )).
In this paper we use this approach to compute the orders of the automorphism groups of groups and Lie algebras G B (F ) resp. g B (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(g B (F ))|; up to a scalar, this formula also gives | Aut(G B (F ))|. We consequently 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. Notwithstanding the fact that our main results are formulated for finite fields, the underpinning structural analysis applies to a larger class of fields, potentially even more general rings.
The following theorem is a condensed summary of the main results of this paper. Throughout we denote, given an elliptic curve E and n ∈ N, by E[n] the n-torsion points of E.
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. Write, moreover, Aut O (E) for the automorphism group of the elliptic curve E and assume that |E[2](F )| = 4. Then there exist groups G 1 (F ), G 2 (F ), and G 3 (F ) such that the following hold: (1) each G i (F ) is a group of order |F | 9 , exponent p, and nilpotency class 2; (2) for each i = 1, 2, 3, there exists Moreover, if δ ∈ F \ {0} is such that E = E δ is given by y 2 = x 3 − δx over F , then We remark that, at least apart from characteristic 3, Theorem 1.1 covers all elliptic curves over Q with j-invariant 1728. It also implies that, for certain values of δ, the function p → | Aut(G i (F p ))| is polynomial on Frobenius sets of primes; see Section 1.5.2 for definitions and details.
In the remainder of the introduction we progressively illustrate some of the paper's ideas. These include explicit constructions, motivation, and broader context. We will prove Theorem 1.1 in Section 5.2.

Finite p-groups from matrices of linear forms
For every prime p, the representation (1.2) gives rise to a finite p-group G 1,1 (F p ) via the following presentation: G 1,1 (F p ) = e 1 , e 2 , e 3 , f 1 , f 2 , f 3 , g 1 , g 2 , g 3 | (1.4) class 2, exponent p, e 1 , e 2 , e 3 and f 1 , f 2 , f 3 abelian, (This ad-hoc definition is a special case of a general construction recalled in Section 2.2.) If p is odd, then G 1,1 (F p ) is a group of order p 9 , exponent p, and nilpotency class 2.
In [dSVL12], du Sautoy and Vaughan-Lee computed the orders of the automorphism groups of the groups G 1,1 (F p ), for primes p > 3. This was a major step towards their aim of showing that the numbers of immediate descendants of these groups of order p 10 and exponent p are not a PORC-function of the primes; see Theorem 1.2 and Sections 1.5.2 and 1.5.3. (The groups G p defined in [dSVL12] can easily be seen to be isomorphic to the groups G 1,1 (F p ).) The purpose of the present paper is twofold: first, to generalize these computations to a larger class of groups (or rather, group schemes); second, to give a conceptual interpretation of the computations in [dSVL12] in terms of Hessian matrices and torsion points of elliptic curves. We reach both aims in Theorem 1.4.
As we now explain, Theorem 1.4 provides new insight, even where it reproduces old results. To see this, note that f 1 = det(B 1,1 ) defines the elliptic curve E 1 : y 2 = x 3 − x over Q. Recall that we denote, given an elliptic curve E over a field F , by E[3] the group of 3-torsion points of E. The collection of its F -rational points E[3](F ) is then isomorphic to a subgroup of Z/(3) × Z/(3). The following is a special case of our Theorem 1.4. Theorem 1.2 (du Sautoy-Vaughan- Lee). Assume that p > 3. Then the following holds: We write µ 4 for the group scheme of 4th roots of unity. We remark that the factor depends only on the equivalence class of p modulo 4. In stark contrast, it follows from the analysis of [dSVL12] that , if p ≡ 1 mod 12 and there exist solutions in F p × F p to y 2 = x 3 − x and x 4 + 6x 2 − 3 = 0, 3, if p ≡ −1 mod 12, 1, otherwise.
(1.5) This case distinction is not constant on primes with fixed residue class modulo any modulus. A concise, explicit description of | Aut(G 1,1 (F p ))|, which is implicit in [dSVL12], is given in [VL12,Sec. 5.2]. Via our Lemma 3.10, one can recover (1.5) from it. In fact, (1.5) uses du Sautoy and Vaughan-Lee's formulation in terms of the solvability of the quartic x 4 + 6x 2 − 3 among the F p -rational points of E 1 . One of the main contributions of the present article is to connect this condition with the structure of the group of 3-torsion points of E 1 , affording an arithmetic interpretation. We remark that the 3-torsion points of E 1 are exactly the flex points of E 1 , i.e. the points which also annihilate Hes(f 1 ) = 8(y 3 3 + 3y 2 1 y 3 − 3y 1 y 2 2 ); see Lemma 3.7.

Generalization 1: further Hessian representations
Our first generalization of Theorem 1.2 is owed to the fact that the Hessian matrix B 1,1 in (1.2) has two natural "siblings".
Indeed, let f ∈ C[y 1 , y 2 , y 3 ] be a homogeneous cubic polynomial defining a smooth projective curve. It is a well-known algebro-geometric fact that the Hessian equation αf = Hes(βf + Hes(f )) (1.6) has exactly three solutions (α, β) ∈ C 2 , yielding pairwise inequivalent linear symmetric determinantal representations of f over C; see also Section 1.4. In fact, any linear symmetric representation of f is equivalent to one arising in this way. For modern accounts of this classical construction, which is presumably due to Hesse [Hes44] The identities 4f 1 = 4 det(B 1,1 ) = det(B 2,1 ) = det(B 3,1 ) are easily verified. Straightforward generalizations of the presentation (1.4) yield, for every odd prime p, groups G i,1 (F p ) of order p 9 , exponent p, and nilpotency class 2. The following generalizes Theorem 1.2.
Theorem 1.3. Assume that p is odd and let i ∈ {1, 2, 3}. Then the following holds: Note that the factor gcd (p − 1, ⌈4/i⌉) is constant equal to 2 for i ∈ {2, 3}. That the cases i = 1, 2, 3 are not entirely symmetric suggests a corresponding asymmetry in the three solutions to the Hessian equation (1.6). The geometric fact ([RT14, Thm. 1(1)]) that they correspond to the nontrivial 2-torsion points of E 1 may help to shed light on this phenomenon.
We also avail ourselves of a general, well-known construction-recalled in detail in Section 2.2-which associates, in particular, to each of the B i,δ a unipotent group scheme . For a finite field F in which δ is nonzero and has a (fixed) square root, we denote by G i,δ (F ) the group of F -rational points of G i,δ . These groups are p-groups of order |F | 9 and nilpotency class 2. We also assume for the rest of the paper that the characteristic of F be odd so that they have exponent p. The groups G i,δ (F ) have a number of alternative descriptions, including one as generalized Heisenberg groups; cf. Section 2.4.1. The following is our first main result.
Theorem 1.4. Let δ ∈ Z and let F be a finite field of characteristic p not dividing 2δ and cardinality p f in which δ has a fixed square root. For i ∈ {1, 2, 3}, the following holds: For any δ ∈ Z \ {0}, the matrices B i,δ (y) defined in (1.9) are inequivalent in the following geometric sense: for any i, j ∈ {1, 2, 3} with i = j, there does not exist U ∈ GL 3 (C) such that U B i,δ U T = B j,δ ; see [PSV12,Prop. 5]. In light of this, it is natural to ask about isomorphisms between groups of the form G i,δ (F ). The next result settles this question, at least for finite prime fields.
Theorem 1.6. Let i, j ∈ {1, 2, 3} and let δ, δ ′ ∈ Z \ {0}, and assume that p is a prime not dividing 2δδ ′ . Then   With a broader outlook, it is of course of interest to consider groups arising from (not necessarily symmetric) determinantal representations of curves or other algebraic varieties. For the time being, however, a general theory connecting geometric invariants of determinantal varieties with algebraic invariants of finite p-groups associated with these varieties' representations eludes us.
Corollary 1.7. Let i ∈ {1, 2, 3} and assume that δ is the fourth power of an integer. Then the function p → | Aut(G i,δ (F p ))| is not PORC. Quasipolynomiality is quite a restrictive property for a counting function. Where it fails, it is natural to look for other arithmetically defined patterns in the variation with the primes. Recall, e.g. from [BMKS19] or [Lag83], that a set of primes is a Frobenius set if it is a finite Boolean combination of sets of primes defined by the solvability of polynomial congruences. A function f : Π → Z is Polynomial On Frobenius Sets (POFS ) if there exist a positive integer N , Frobenius sets Π 1 , . . . , Π N partitioning Π, and polynomials f 1 , . . . , f N ∈ Z[T ] such that the following holds: Our Theorem 1.4 implies, for instance, the following.
In fact, for δ = 1 one may take N = 4 for i = 1 and N = 3 for i ∈ {2, 3}. Corollary 1.8 invites a comparison of Theorem 1.4 with a result by Bardestani, Mallahi-Karai, and Salmasian. Indeed, [BMKS19, Thm. 2.4] establishes-in notation closer to the current paper-that, for a unipotent group scheme G defined over Q, the faithful dimension of the p-groups G(F p ) (viz. the smallest n such that G(F p ) embeds into GL n (C)) defines a POFS function. For a number of recent related quasipolynomiality results, see [EVL20].

Automorphism groups and immediate descendants
Du Sautoy and Vaughan-Lee embedded their discussion of the automorphism groups of the groups G 1,1 (F p ) in a study of the immediate descendants of order p 10 and exponent p of these groups of order p 9 . In fact, their paper's main result is the statement that the numbers of these descendants is not PORC as a function of p. Theorem 1.2 allows us to give a compact, conceptual formula for these numbers. Let n 1,1 (p) denote the number of immediate descendants of G 1,1 (F p ) of order p 10 and exponent p. Set, moreover, e(p) = |E 1 [3](F p )| and m(p) = gcd(p − 1, 4). Corollary 1.9 (du Sautoy-Vaughan- Lee). Assume that p > 3. Then the following holds: We hope to come back to the interesting question of how to generalize and conceptualize this work to the groups G i,δ (F ) for δ ∈ Z \ {0} and i ∈ {1, 2, 3} in a future paper.

Further examples and related work
In [Lee16], Lee constructed an 8-dimensional group scheme G, via a presentation akin to (1.4), and proved a result which is similar to Theorem 1.2. In particular, he showed that both the orders of Aut(G(F p )) and the numbers of immediate descendants of G(F p ) of order p 9 and exponent p vary with p in a nonquasipolynomial way. More precisely, these numbers depend on the splitting behaviour of the polynomial x 3 − 2 over F p or, equivalently, on the realisability over F p of permutations of 3 specific, globally defined points in P 1 .
In [VL18], Vaughan-Lee proved a similar result about p-groups arising from a parametrized family of 7-dimensional unipotent group schemes of nilpotency class 3. He proved that the orders of the automorphism groups of these p-groups, which feature two integral parameters x and y, depend on the splitting behaviour modulo p of the polynomial It is natural to try and further expand the range of computations of automorphisms of p-groups obtained as F -points of finite-dimensional unipotent group schemes, where F is a finite field.
Arithmetic properties of finite p-groups arising as groups of F -rational points of group schemes defined in terms of matrices of linear forms are also a common theme of [BI04, OV15, Ros20, RV19] (with a view towards the enumeration of conjugacy classes) and [BMKS19] (with a view towards faithful dimensions; cf. also Section 1.5.2).
Slight variations of the group schemes G 1,δ were highlighted by du Sautoy in the study of the (normal) subgroup growth of the groups of Z-rational points of these group schemes. Indeed, he showed in [dS02] that the local normal zeta functions of these finitely generated torsion-free nilpotent groups depend essentially on the numbers of F p -rational points of the reductions of the curves E δ modulo p; see also [Vol04,Vol05] for explicit formulae and generalizations.
The groups we consider arise via a general construction of nilpotent groups and Lie algebras from symmetric forms. Automorphism groups of such groups and algebras have been studied, e.g. in [GS08,Wil17,BMW17]. In fact, our Theorem 1.4 may be seen as an explicit version of [GS08, Thm. 7.2] and [Wil17, Thm. 9.4]. The arithmetic point of view taken in the current paper seems to be new, however.

Organization and notation
The paper is structured as follows. In Section 2 we recall the well-known general construction yielding the nilpotent groups and Lie algebras considered in this paper and set up the notation that will be used throughout the paper. In Section 3 we gather some elementary results about automorphisms of elliptic curves. In Section 4 we collect a number of structural results of the groups in question necessary to determine their automorphism groups. We apply these, in combination with the results of the previous sections, to give a proof of the paper's main results and their corollaries in Section 5.
The notation we use is mainly standard. Throughout, k denotes a number field, with ring of integers O k . The localization of O k with respect to the powers of δ ∈ Z is denoted by R = O k,δ . By K we denote a field with an R-algebra structure, in practice nearly always an extension of k or a residue field of a nonzero prime ideal of O k . By F we denote a finite field.

Groups and Lie algebras from (symmetric) forms
In this section we work with groups and Lie algebras arising from symmetric matrices of linear forms via a classical construction which we review in Sections 2.1 and 2.2. Our reasons to restrict our attention to symmetric matrices, rather than discuss more general settings, will become apparent in Section 2.3; see also Remark 2.1.

Global setup
The following is the setup for the whole paper. Let k be a number field with ring of integers O k . For a nonzero integer δ, we write R = O k,δ for the localization of O k at the set of δ-powers. Let further d be a positive integer and let U , W , and T be free R-modules of rank d. Let y = (y 1 , . . . , y d ) be a vector of independent variables and let B be a symmetric matrix whose entries are linear homogeneous polynomials (which may be 0) over R, i.e.
. . , f d ), and T = (g 1 , . . . , g d ) be R-bases of U , W , and T respectively, allowing us to identify each U , W , and T with R d . Let, moreover, be the R-bilinear map induced by B with respect to the given bases. Throughout we write ⊗ to denote ⊗ R . We denote byφ : the homomorphism that is given by the universal property of tensor products. By slight abuse of notation, we will use the bar notation ( ) both for the map u j f j and its inverse W → U . We remark that the symmetry of B yields, for u ∈ U and w ∈ W , for the free R-modules of ranks 2d respectively 3d. Our notation will reflect that we consider the summands of these direct sums as subsets.
Throughout, K is a field with an R-algebra structure. (In practice, K may be, for instance, an extension of the number field k or one of various residue fields of nonzero prime ideals of O k coprime to δ.) By slight abuse of notation we use, in the sequel, the above R-linear notation also for the corresponding K-linear objects obtained from taking tensor products over R. For example, L will also denote the K-vector space L ⊗ R K.

Groups and Lie algebras
The data (K, B, φ) gives the K-vector space L a group structure. Indeed, with (1) the identity element is (0, 0, 0), (2) the inverse of the element (u, w, t) is (u, w, t) −1 = (−u, −w, −t + φ(u, w)), (3) the commutator of any two elements (u, w, t) and (u ′ , w ′ , t ′ ) is (2.5) The data (K, B, φ) also endows L with the structure of a graded K-Lie algebra. Indeed, with ) is a nilpotent K-Lie algebra of nilpotency class at most two, viz. the K-rational points of an R-defined Lie algebra scheme g B (abelian if and only if B = 0). Note that the Lie bracket (2.6) coincides with the group commutator (2.5).
The property of B being symmetric is not an isomorphism invariant of G B (K). Indeed, linear changes of coordinates on U and W preserve the isomorphism type of G B (K) but result in a transformation B → P T BQ for some P, Q ∈ GL d (K). For computations, however, the symmetric setting proved much more conventient to work with. Clearly, symmetric coordinate changes (i.e. P = Q) preserve the matrix's symmetry.
We focus on groups G B (K) where B is a symmetric 3 × 3 determinantal representation of a planar elliptic curve E. As discussed in Section 1.3, there are, over the algebraic closure of K, three inequivalent such representations B 1 , B 2 , B 3 , where inequivalent means that they belong to three distinct orbits under the standard action of GL 3 (K) 2 on 3 × 3 matrices of linear forms. In particular, equivalent representations yield isomorphic groups, but, as our Theorem 1.6 shows, inequivalent representations might yield isomorphic groups.
In the case that K = F is a finite field of odd characteristic p, the groups G B (F ) are the finite p-groups associated with the Lie algebras g B (F ) by means of the classical Baer correspondence ( [Bae38]). It was anticipated, in the case at hand, in Brahana's work ( [Bra35]) and extended in the much more general (and better known) Lazard correspondence; [Khu98, Exa. 10.24]. It implies, in particular, that | Aut(G B (F p ))| = | Aut(g B (F p ))| and, more gener- To lighten notation we will, in the sequel, not always notationally distinguish between the K-Lie algebra scheme g = g B and its K-rational points. We trust that the respective contexts will prevent misunderstandings.

Groups of Lie algebra automorphisms
By Aut K (g) = Aut(g) we denote the automorphism group of the K-Lie algebra g. Define, additionally, We remark that Aut V (g) is nothing but the centralizer C Aut(g) (V ) considered with respect to the action of Aut(g) on the Grassmannian of L. Our choice of bases allows us to identify Aut(g) with a subgroup of GL 3d (K). We may thus view each element α of Aut(g) as a matrix of the form . The subgroup Aut V (g) of Aut(g) comprises those matrices with entries C = D = 0 in (2.7). It is easy to show that We also observe that every element of Aut f V (g) is of the form and belongs to Aut = V (g) if and only if, in addition, A U = A W . We define the map ψ : GL 2 (K) → Aut(g) by Lemma 2.2. The map ψ is an injective homomorphism of groups.
Proof. We show that ψ is well-defined. Addition in the Lie algebra is clearly respected, so it suffices to show that the Lie brackets are respected, too. Let (u, w, t) and (u ′ , w ′ , t ′ ) in g and To show that ψ is an injective homomorphism is a routine check.
Let u, u ′ ∈ U and let w ∈ W . Then one easily computes In particular, U is an abelian subalgebra of g and, by symmetry, so is W . As a consequence of this fact and Lemma 2.2, for each M ∈ GL 2 (K), the subspaces ψ(M )(U ) and are related in the following way: (1) =⇒ (2) ⇐⇒ (3).
Proof. (1) ⇒ (2): A given element γ ∈ GL(T ) corresponds, via the identification T ∼ = K d induced by the basis T , to a matrix A T ∈ GL d (K). We denote by γ(y 1 ), . . . , γ(y d ) ∈ K[y 1 , . . . , y d ] the images of the variables y 1 , . . . , y d under A T , acting on K[y 1 , . . . , y d ] in the standard way. Writing B = B(y 1 , . . . , y d ), we analogously define (2) ⇔ (3): Clearly X is a complement of U in V . We obtain that X is an abelian subalgebra of g if and only if, for any choice of w, w ′ ∈ W , the element [w + wD T , w ′ + w ′ D T ] is trivial. This happens if and only if, for all w, w ′ ∈ W , the following equalities hold: As a consequence, X is abelian if and only if D T B = BD.
Lemma 2.4. Assume that all d-dimensional abelian subalgebras of g that are contained in V are of the form ψ(M )(U ) for some M ∈ GL 2 (K). Then the following hold: (1) Aut V (g) = ψ(GL 2 (K)) Aut f V (g).
(2): Let X be a d-dimensional abelian subalgebra of g that is contained in V . It follows from the assumptions that either X = U or there exists λ ∈ K such that In other words, the complements of U in V that are also abelian subalgebras of g are parametrized by the scalar matrices in is actually a subgroup of Aut f V (g), which is easily seen to be normal. Moreover, the intersection N ∩ Aut = V (g) is trivial, by definition of Aut = V (g). In order to prove Theorem 1.4 in Section 5.1, we will make use of Lemma 2.4 via Proposition 4.10.
Lemma 2.5. Let A ∈ GL d (K) and assume that the image of φ spans T . Then the following are equivalent: Proof. (1) ⇒ (2): Assume that (A ⊗ A)(kerφ) ⊆ kerφ. Then A ⊗ A induces an isomorphism of T in the following way. For j ∈ {1, . . . , d}, choose an element v j ∈ U ⊗ W such that φ(v j ) = g j . Define A T : T → T to be the K-linear homomorphism that is induced by for j = 1, . . . , d. The map A T is well-defined since A ⊗ A stabilizes the kernel ofφ. Moreover, the following diagram is commutative. ) is surjective, A T is surjective and thus an isomorphism. In particular, diag(A, A, A T ) : U ⊕ W ⊕ T → U ⊕ W ⊕ T is an automorphism of the K-Lie algebra g.
(2) ⇒ (1): Let u ∈ U and let w ∈ W , which we also regard as elements of g. Then holds and so we are done.
Remark 2.6. We exploit the symmetry of B, for example, in the following way. Let v = d i,j=1 a ij e i ⊗ f j be an element of U ⊗ W and set C = A T BA, which is by definition a symmetric matrix. Then we have the identities a ij e i Cf T j .
). We will put this to use in Section 5.1.

Alternative descriptions
The groups G B (K) defined in Section 2.2 have alternative descriptions as follows.

Heisenberg groups
Let H be the group scheme of upper unitriangular 3 × 3-matrices. Then G B (K) is equal to H(A), where A = Kg 1 ⊕ · · · ⊕ Kg d is the K-algebra given by setting g r · g s = (B(g 1 , . . . , g d )) rs for r, s ∈ {1, . . . , d}. Note that the algebra A is commutative (as the matrix B is symmetric) but in general not associative. The group H(A) is nilpotent of class at most 2, and abelian if and only if B = 0. If A is associative, then H(A) is called a Verardi group; c.f. [GS08].
We may recover the Lie algebra g B (K) defined in Section 2.2 from the group G B (K). Indeed, we find that g B (K) is isomorphic to the graded Lie algebra

Central extensions and cohomology
The group G B (K) is a central extension of the abelian group V = U ⊕ W by T and corresponding to the 2-cocycle where the action of V on T is taken to be trivial. This classical construction can be found, for example, in [Bro82, Ch. IV]. The equivalence classes of central extensions of V by T , in the sense of [Bro82, Ch. IV.1], are in 1-to-1 correspondence with the elements of H 2 (V, T ). Equivalence being a stronger notion than isomorphism, the image of φ V in H 2 (V, T ) will generally not suffice to determine the isomorphism type of G B (K).
3 Automorphisms and torsion points of elliptic curves 3.1 General notation and standard facts Let K be a field and let E be an elliptic curve over K with point at infinity O. For a positive integer n, we write E[n] for the n-torsion subgroup of E and E(K) resp. E[n](K) for the K-rational points of E resp. E[n]. We define, additionally, containing as a subgroup the automorphisms of E as elliptic curve, or invertible isogenies, for more information, see for example [Sil09, Chap. III.4]. Except for the notation for the automorphism groups, we will refer to results from and notation used in [Sil09, Chap. III]. We warn the reader that in [Sil09] the notation Aut(E) denotes what we defined as Aut O (E).
In Section 3.2 we determine, for specific elliptic curves, which of their endomorphisms are induced by (or lift to) linear transformations of the plane. To this end we define We write, additionally, c E for the natural homomorphism Remark 3.1.
Any endomorphism ϕ of E as a projective curve can be written as a composition ϕ = τ • α, where τ is a translation and α is an isogeny E → E. In other words, each element of Aut(E) can be written as the composition of a translation with an element of Aut O (E). There thus exists an isomorphism Lemma 3.2. Assume that K is algebraically closed. Then there exists a subset U of K 3 of cardinality 4 such that the projective image of U is contained in E(K) and any 3 elements of U form a basis of K 3 .
Proof. Let Q 1 , . . . , Q 4 ∈ E(K) be four points of coprime orders |Q i | with respect to addition in E. Then, for each (i, j) ∈ {1, 2, 3, 4} 2 , the order |Q i + Q j | is the least common multiple of |Q i | and |Q j |. Hence the definition of the group law implies that no three points of {Q 1 , Q 2 , Q 3 , Q 4 } lie on the same projective line. Any collection of non-zero lifts of the Q i to elements of K 3 will do. Proof. Without loss of generality we assume that K is algebraically closed. Let ϕ ∈ GL 3 (K) be such that ϕ ∈ ker c E . Let, moreover,Ẽ be the affine variety corresponding to E in K 3 . Since ϕ induces the identity on E, every element ofẼ(K) is an eigenvector of ϕ. Then, Lemma 3.2 yields not only that ϕ is diagonalisable, but also that all eigenvectors have the same eigenvalue. In particular, there exists λ ∈ K × such that ϕ equals scalar multiplication by λ and thus ϕ is trivial.

A parametrized family of elliptic curves
Let δ = 0 be an integer, let K be a field of characteristic not dividing 2δ, and let E δ be the elliptic curve defined over K by The projectivisation of E δ , obtained by setting z = δ −1 , is given by and has point at infinity equal to O = (0 : 1 : 0). The j-invariant of E δ being equal to 1728, the automorphism group Aut O (E δ ) consists of all maps of the form (x, y) → (ω 2 x + ρ, ω 3 y), where (ω, ρ) satisfies ω 4 = 1 and δρ 3 = ρ with ρ = 0 only if char(K) = 3; see [Sil09, Thm. III.10.1, Prop. A.1.2]. To lighten the notation, we will write X δ for X E δ and c δ for c E δ .
Proof. Proving that A induces α is straightforward; uniqueness follows from Lemma 3.3.
In the next lemma, let δ these are the nontrivial 2-torsion points of E δ over K. For a point Q of E δ we denote, additionally, by τ Q the translation τ Q : E δ → E δ , P → P + Q.
As pointed out in Remark 3.1, the map τ Q is an element of Aut(E δ ).
Proof. Let A ∈ GL 3 (K) be such that c δ (A) = τ Q . Without loss of generality, assume that K = K. For a contradiction we assume, moreover, that Q = P 1 ; the other cases are analogous. Since P 1 is an element of order 2, we find that τ Q (O) = P 1 , τ Q (P 1 ) = O, τ Q (P 2 ) = P 3 , and τ Q (P 3 ) = P 2 .
We now give a geometric interpretation of the quartic polynomial featuring in Lemma 3.6.
Lemma 3.9. Let Q = (a, b) be a point of E δ [3] and assume that A ∈ GL 3 (K) is such that c δ (A) = τ Q . Then there exists ν ∈ K × such that, up to a scalar, Proof. From the addition formulas we derive Moving to affine coordinates (u 1 , u 2 , u 3 ) we may find λ, ν, γ ∈ K × such that This implies that Multiplying A by 2δab gives the claim.
We close this section with an observation in the special case when δ = ε 4 for some ε ∈ K. As explained in Section 1.2, it is used to establish (1.5). To contextualize this situation, we remark that, when δ ′ ∈ Z \ {0} and 6δ ′ is not divisible by char(K), then the elliptic curves E δ and E δ ′ are isomorphic over K if and only if there exists some ε ∈ K such that δ = δ ′ ε 4 ; cf. [Sil09, Ch. III.1]. Indeed, an isomorphism E δ → E δ ′ is given by the invertible isogeny (x, y) → (ε 2 x, ε 3 y).
Lemma 3.10. Assume that char(K) = 3 and that δ = ε 4 for some ε ∈ K. Define, moreover, Then there exists bijections Proof. Lemma 3.7 yields a bijection E δ [3](K) \ {O} → S 1,δ . A bijection S 1,δ → S 1 is given by the restriction of the isogeny (a, b) → (ε 2 a, ε 3 b). For the last arrow note that, with S 2 = {(a, b) ∈ K 2 | 1 − a 2 + ab 2 = 0 and a 4 + 6a 2 − 3 = 0} and the fact that char(K) = 3, the maps are well-defined and mutually inverse bijections. The sets S 2 and S 0 are in bijection as, for a fixed element a ∈ K with a 3 + 6a 2 − 3 = 0, we find that a = 0 and thus the solutions to the equation b 2 = a 3 − a are in bijection with the solutions of the equation b 2 = a 2 −1 a .

Degeneracy loci and automorphisms of p-groups
This section brings together the constructions from Section 2 and the facts about automorphisms of elliptic curves from Section 3. In the case that the determinant of the matrix of linear forms B determines a elliptic curve E, we define an explicit homomorphism In the case that, in addition, B is a Hessian matrix and F is a finite field of odd characteristic, the current section's main result Corollary 4.14 yields a formula for | Aut(g B (F ))| in terms of the size of the image of c B . To prove Theorem 1.4 in Section 5.1 we are just left with determining this image size explicitly. Throughout Section 4 we continue to use the notation introduced in Section 2.1. Recall, in particular, the definition (2.1) of the matrix of linear forms B(y) ∈ Mat d (R[y 1 , . . . , y d ]) in terms of "structure constants" B Remark 4.2. The matrices B(y) and B • (x) are, in a precise sense, dual to one another. Indeed, as we pointed out in Section 2.4.2, the matrix B characterizes a module representation θ • : T * → Hom(W, U * ). The dual matrix B • then characterizes the module representation θ = (θ • ) • : U → Hom(W, T ); see [Ros20, § 4.1] and Definition 4.3. In particular, they satisfy (4.1) The matrices B(y) and B • (x) are also closely related to the "commutator matrices" defined in [OV15, Def. 2.1]. In this paper's notation, we find that, for vectors of algebraically independent variables Y = (Y 1 , . . . , Y d ) and X = (X 1 , X 2 ) = (X 1 , . . . , X d , X d+1 , . . . , X 2d ), .  φ(u, w)).
(2): Let v ∈ ker B(u). Then the assumption implies that B(u)Dv T = D T B(u)v T = 0.
We next observe that matrices of linear forms that are, as the ones in (1.9), defined as Hessian matrices, have a remarkable self-duality property.
Write v ∈ V as v = u + w with u ∈ U and w ∈ W . One checks easily that cf. also Lemma 4.4(1) and Remark 4.2. In particular, if v = u is an element of U and PV B • is smooth, then dim K C V (u) = 2d − rk B • (u) and so, by Lemma 4.8, the following holds: otherwise. (4.2) If, additionally, d = 3 and B = B • , as in the situation described in Lemma 4.5, then we find that dim K C V (u) = 6 − rk B(u) and hence (4.3) We remark that this applies, in particular, to the matrices B i,δ defined in (1.9)

Implications for automorphism groups of p-groups
For the rest of the section we assume that d = 3. We mainly describe, in Proposition 4.10, the 3-dimensional abelian subalgebras of g that are contained in V . This allows us to apply Lemma 2.4, leading to a refined formula for | Aut(g B (F ))| in the case that F is a finite field and PV B and PV B • are elliptic curves; see Corollary 4.14.
Proposition 4.10. Assume that PV B and PV B • are elliptic curves. Then the 3-dimensional abelian subalgebras of g that are contained in V are exactly those of the form ψ(M )(U ) for some M ∈ GL 2 (K).
Proof. Let X be a 3-dimensional abelian subalgebra of g that is contained in V . Assume first that X ∩U = {0}. We claim that X = U and assume, for a contradiction, that X = U . Then there exists u ∈ X ∩ U with centralizer C V (u) of dimension at least 4 and such that X is contained in C V (u). Then (4.2) implies that dim K C V (u) = 4 or, equivalently, that the kernel of Φ u has dimension 1. Let w ∈ W be such that ker Φ u = Kw and define Y = U ⊕ ker Φ u so that Y has dimension 4. Since X and U do not coincide, it follows that Y = X + U and X ∩ U has dimension 2. We let t ∈ X be such that X = (X ∩ U ) ⊕ Kt and denote by π U : V → U resp. π W : V → W the natural projections. Since t belongs to Y , we have π W (t) = λt for some λ ∈ K × . It follows that 0 = φ(X ∩ U, t) = φ(X ∩ U, π U (t) + π W (t)) = φ(X ∩ U, π W (t)) = λφ(X ∩ U, w), which implies that Y is an abelian subalgebra of g of dimension 4. As a consequence, w is a central element and so, by Lemma 4.4(1), the rank of B • (w) is zero; contradiction to Lemma 4.8. To conclude, in this case one can take M = Id 2 to get that X = ψ(M )(U ) = U . Assume now that X ∩ U = {0}. Then X is a complement of U in V , as W is, and thus there exists D ∈ Mat 3 (K) such that X = {w + wD T | w ∈ W }. Fix such a D. By Lemma 2.3, it satisfies D T B = BD. We claim that D is a scalar matrix, i.e. D = λ Id 3 for some λ ∈ K. For this we may, without loss of generality, assume that K is algebraically closed. Indeed, solving D T B = BD over K reduces to solving a system of 9 linear equations in 9 indeterminates and, if D T B = BD implies that D is scalar over an algebraic closure of K, then the same holds over K. Let U be a subset of V B • (U ) of cardinality 4 such that any 3 of its elements form a basis of U ; such U exist by Lemma 3.2. By (4.1) there exists, for each u ∈ U, an element w u ∈ W \ {0}, unique up to scalar multiplication, such that In particular, combining Lemma 4.8 and (4.1) yields a well-defined bijection with inverse P(Kw) → P(ker B(w)). It follows from Lemma 4.4(2) and Lemma 4.8 that each u ∈ U is an eigenvector with respect to D with eigenvalue λ u ∈ K, say. The fact that any three elements of U generate U yields the existence of a λ ∈ K such that for each u ∈ U one has λ = λ u ; in particular, the matrix D is equal to λ Id 3 . For such λ, defining M = 1 λ 0 1 yields that X = ψ(M )(U ).
Corollary 4.11. Let F be a finite field of odd characteristic and assume that PV B and PV B • are elliptic curves over F . Then the following holds: Proof. Write g for g B (F ) and let C g denote the centroid of g, as defined in [Wil17, § 1.1]. Using Proposition 4.10 and extending the arguments of Lemmas 2.3 and Lemma 2.4, it is not difficult to see that C g is the collection of all the scalar multiplications on g by elements of F and so C g ∼ = F . As a consequence of [Wil17, Thm. 1.2(D)] (cf. also [Wil17,§5]) we obtain an exact sequence As mentioned in Section 2, Proposition 4.10 allows us to apply Lemma 2.4 in the proof of Theorem 1.4. This lemma reduces the determination of the order of the automorphism group of g B to the analysis of the structure of the subgroup Aut = V (g B ) of Aut(g B ). In the next proposition, let π : Proposition 4.13. Assume that PV B • is an elliptic curve. Then the map is a well-defined homomorphism of groups and satisfies c B = c PV • B • π. Proof. The elements of Aut = V (g B ) stabilize the abelian subalgebra U and, being Lie algebra homomorphisms, respect the centralizer dimensions dim C V (u) of the elements u ∈ U . More concretely, if ϕ belongs to Aut = V (g B ), then, thanks to Remark 4.9, the following hold: In The idea of appealing to the subgroup Aut = V (g) in order to determine the structure of Aut(g) was already pursued in [dSVL12]. This notwithstanding, the map c B holds the key to the realization of the added value brought about by our geometric point of view. Indeed, the determination of the image of c B plays a decisive role in the proof of Theorem 1.4 by means of the following corollary.
Corollary 4.14. Let F be a finite field of odd characteristic and assume that PV B and PV B • are elliptic curves over F . Then the following hold: Proof. The combination of (2.8), Proposition 4.10, and Lemma 2.4(4) yields that The claim follows as Lemmas 3.3 and 2.4(3) imply that ker c B = {λ Id 9 | λ ∈ F × } ∼ = F × .

Proofs of the main results and their corollaries
We prove Theorem 1.4 and its Corollaries 1.7-1.9 in Section 5.1 and Theorem 1.6 in Section 5.2, where we also prove Theorem 1.1.

Automorphisms
We continue to use the setup from Section 3.2 and combine it with that from Section 2.1.
Recall that δ is a nonzero integer and that K is a field of characteristic not dividing 2δ. Assume further that K contains a fixed square root δ 1 2 of δ. For i ∈ {1, 2, 3}, let B i,δ be as in (1.9), and denote by φ i,δ and φ i,δ , respectively, the associated bilinear and linear maps defined in (2.2) resp. (2.3). The image of φ i,δ spans T so φ i,δ is surjective.
Write G i,δ = G B i,δ and g i,δ = g B i,δ , respectively, for the group and the Lie algebra (schemes) associated with the data (K, B i,δ , φ i,δ ) in Section 2.2. We recall that, if K = F is a finite field of order q and odd characteristic p, then the finite p-group G i,δ (F ) has exponent p, nilpotency class 2, and order q 9 , while the F -Lie algebra g i,δ has F -dimension 9 and nilpotency class 2. We are looking to compute the order of Aut(g i,δ (F )). For this, we will use Corollary 4.14. Indeed, the matrix B i,δ is Hessian and therefore satisfies B i,δ = B • i,δ ; see Lemma 4.5. Observe that PV B i,δ is identified with the elliptic curve E δ via the projectivisation (3.1). Let c B i,δ : Aut = V (g i,δ ) → Aut(E δ ) be the homomorphism from Proposition 4.13. By Proposition 3.8, its image is isomorphic to a subgroup of E δ [3] ⋊ Aut O (E δ ), which leads us to consider the homomorphism (5.1) Corollary 4.14 reduces the proof of Theorem 1.4 to the explicit determination of the image size of c i,δ . To this end, we will make use of the following specific version of Lemma 2.5.
Proof. Consider the following supersets of K * i,δ of K i,δ in U ⊗ W : The kernel of φ i,δ is spanned by K i,δ over K. It follows, however, from Remark 2.6 that to check whether (A ⊗ A)(ker φ i,δ ) is contained in ker φ i,δ it suffices to check if (A ⊗ A)(K * i,δ ) is annihilated by φ i,δ . Indeed, the elements of K i,δ \ K * i,δ are equal to the negatives of their respective duals or duals to members of K * i,δ . To conclude we apply Lemma 2.5.
Proof. Up to scalar multiplication, Lemma 3.9 yields that a necessary condition for the existence of such a pair (A, A T ) is the existence of ν ∈ K × such that We set ν = −δ/(δa 2 + 1) and claim that, for this matrix A = A(ν), there exists A T = A T (ν) such that c i,δ (diag(A, A, A T )) is equal to (Q, id E ). Indeed, checking (e.g. with SageMath [The19]) identities involving the matrix C = A T B i,δ A defined in Remark 2.6, one shows that (A ⊗ A)(K * i,δ ) ⊆ ker φ i,δ . We conclude with Lemma 5.1. Corollary 5.4. Let F be a finite field of characteristic not dividing 2δ in which δ has a fixed square root. Then the following holds: Proof. Combining Lemmas 5.2 and 5.3 allows us to describe the image of the homomorphism c i,δ (cf. (5.1)) and hence, by Corollary 4.14, the order of Aut(g i,δ (F )).
To prove Theorem 1.4 it now suffices to note that | Aut(G i,δ (F ))| = | Aut Fp (g i,δ (F ))| by the Bear correspondence. We may thus conclude by combining Corllary 4.11 and and Lemma 5.4.
We conclude by proving Corollaries 1.7-1.9. To this end, let p be a prime not dividing 6δ and let n 1,1 (p) denote the number of immediate descendants of G 1,1 (F p ) of order p 10 and exponent p. In [dSVL12,Sec. 4] it is shown that neither of the two functions p −→ | Aut(G 1,1 (F p ))|, p −→ n 1,1 (p) are PORC, the second being so as a consequence of the first. Combining Theorem 1.3 with Lemma 3.7, this amounts to saying that the function Π → Z, p → |E 1 [3](F p )| is not constant on residue classes modulo a fixed integer. Corollary 1.7 now follows from Theorem 1.4 and Lemma 3.10. To prove Corollary 1.8, observe that the function p → |E[3](F p )| is constant on Frobenius sets for any elliptic curve E defined over Q. To prove Corollary 1.9 one may proceed as in [dSVL12, Sec. 11], i.e. by counting orbits of the induced action of im c 1,1 on F 3 p by means of Burnside's lemma. One checks easily that our formula for the descendants matches the values listed in [VL12, Sec. 5].

Isomorphisms
While we focussed so far on automorphisms of groups and Lie algebras of the form described in Section 2.2, we now turn to isomorphisms between such objects. The section's main aim is to prove Theorem 1.6. To this end, we state and work with a number of results that have close counterparts in Section 2. Their proofs being entirely analogous, we omit most of them. We also restrict to the case that char(K) = 3, to lighten notation.
The following is a variation of Lemma 2.3. We omit the analogous proof.
Proof. Let α be as in Lemma 5.5 and write α = diag(A U , A W , A T ) for matrices A U , A W , A T in GL 3 (K). Let, moreover, α U , α W denote the projective curve isomorphisms E δ → E δ ′ induced by A U resp. A W .
The following is a variation of Lemma 2.5. We omit the analogous proof.
Lemma 5.9. Let β = diag(A, A, A T ) and β ′ = diag(A ′ , A ′ , A ′ T ) be Lie algebra isomorphisms g i,δ (K) → g j,δ ′ (K) such that A and A ′ induce the same invertible isogeny E δ → E δ ′ . Then A and A ′ are equal in PGL 3 (K). belong to kerφ 3,δ ′ . Applyingφ 3,δ ′ and using the fact that the basis elements g