• Corpus ID: 232427866

A moduli interpretation of untwisted binary cubic forms

@inproceedings{Kulkarni2021AMI,
  title={A moduli interpretation of untwisted binary cubic forms},
  author={Rajesh S. Kulkarni and Charlotte Ure},
  year={2021}
}

• Rajesh S. Kulkarni, C. Ure
• Published 30 March 2021
• Mathematics

We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural GL2-action on the variables. In particular, we show that these GL2 orbits are in bijection with pairs of j-invariant 0 elliptic curves together with 3-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence. Introduction The subject of homogeneous forms has fascinated mathematicians for centuries as it relates various branches such as algebraic geometry and number theory. Our particular interest is in binary forms which have been studied in great detail. For example, the invariants and covariants of binary forms of degree 3 and 4 were classically described by Schur et al. [29]. Eisenstein also investigated binary cubic forms (over Z) by constructing a finite map to the space of quadratic forms. He counted the number of elements in a fiber as a step towards counting the number of representations of a positive integer as a given cubic form [10]. Binary forms of higher degree were studied, for example, by Franklin and Sylvester [33]. Binary cubic forms over Z have been of great interest in number theory in relation to counting cubic extensions of Q. The idea is as follows: A cubic field extension of Q is generated by attaching a root of a polynomial f(x) ∈ Q[x]. However, different polynomials may generate the same field. For example, the leading coefficient plays no role in generating the cubic field and the polynomial obtained from a generating polynomial via a linear transformation of the variable also generates the same field. In this vein, a result of Davenport [8, 9] gives an asymptotic expression for the number of cubic fields as the discriminant approaches infinity. In recent times, binary cubic forms over Z were studied by various authors: Bhargava-Gross [2], Gan-Gross-Savin [11], and Wood [38]. The main goal of these works is to use the moduli space of binary cubic forms to count the number of cubic field extensions of Q. These works begin with the parameter space of all nondegenerate binary cubic forms, and then consider the quotient by GL2. The action of GL2 is twisted by the determinant to account for the fact that different polynomials may generate the same cubic field. In this article, we investigate the natural question: Can we describe the moduli space obtained as the quotient of the space U of (nondegenerate) binary cubic forms under the natural untwisted action of GL2? This raises other related questions: Does the family of genus 1 curves over U descend to this quotient? What about the associated family of Jacobians of these curves? We study these questions over any base field of characteristic different from 2 and 3 containing a primitive third root of unity ω. It turns out that there is a unifying theme that ties up these aspects. There is a noncommutative algebra, called the generic Clifford algebra that acts as the universal algebra where the forms are linearized. The generic Clifford algebra An,d of degree d in n variables was introduced by 1 Chan, Young, and Zhang in [5]. It is the quotient of the tensor algebra TV given by An,d = TV 〈[ud, v] : u, v ∈ V 〉 , where V is a k-vector space of dimension n and [ u, v ] denotes the commutator. The algebraic (ring-theoretic) properties of this algebra have been studied in some cases. For example, when d = 2, this algebra is a connected, graded, Artin-Schelter regular, Auslander regular, Cohen Macauly noetherian domain [5]. Our interest is in the binary cubic case (n = 2, d = 3). The algebra A = A2,3 is called the binary cubic generic Clifford algebra. It is easy to see that the algebra A can be described by generators and relations as A = k 〈 x, y : xy = yx, xy = yx, xy + (xy) = yx + (yx) 〉 . This algebra has been studied in the context of the classification of Artin-Schelter regular algebras. Wang and Wu showed that it is an Artin-Schelter regular, strongly noetherian, Auslander regular, Cohen Macauly algebra of global dimension 5 [36]. Its representation theoretic properties were explored by Wang and Wang in [35]. We explore other algebraic properties of A in Section 1. In particular, we show that the center of A is isomorphic to the affine coordinate ring of a relative projective curve over A that is elliptic over a Zariski open dense subset U . This subset U corresponds to the set of nondegenerate binary cubic forms over k. One of our main results is the following combination of Theorem 2.1 and Theorem 2.13 that relates the algebra A to the moduli problem discussed above: Theorem 0.1. The binary cubic generic Clifford algebra A defines a Brauer class over a relative elliptic curve over U . The associated Brauer class is not trivial for any base field k. We will now describe how the binary cubic generic Clifford algebra relates to the GL2 action on binary cubic forms described above. The natural GL2-action on the two dimensional vector space V extends to the algebra A. In particular, it induces an action on the center Z(A). This action is compatible with the GL2-action on the affine space A and it preserves the open subset U of nondegenerate binary cubic forms (see Section 3). Combining the binary cubic generic Clifford algebra with this GL2-action on nondegenerate binary cubic forms, we prove that the associated Brauer class descends to the quotient stack Proposition 3.14. For our investigation of the moduli space of binary cubic forms, we use yet another algebra called the generalized Clifford algebra associated to a form f [6, 27, 28]. Let f be a binary homogeneous form of degree 3 over a field k. The generalized Clifford algebra Af associated to f is the quotient of the tensor algebra Af = TV 〈v3 − f(v) : v ∈ V 〉 . This is a natural generalization of the classical Clifford algebra associated to a quadratic form. If f is given as f(u, v) = au+3buv+3cuv+dv, we can write the algebra Af using generators and relations as Af = k 〈 x, y : x = a, xy + xyx+ yx = 3b, xy + yxy + yx = 3c, y = d 〉 . This algebra is an Azumaya algebra over its center, the complement of zero in the Jacobian of w − f(u, v) [15, 19]. Furthermore, the associated Brauer class extends to the Jacobian and is trivial if and only if the curve w3− f(u, v) admits a k-rational point [16]. These works also give 2 an explicit equation for the Jacobian as s = γ + 1 4 ∆, where ∆ is the discriminant of the form f . We continue with the discussion of the results of this article. We show that the action of GL2 on the set of nondegenerate binary cubic forms extends to an action on the center Z(A) of A (see Section 3). In Theorem 3.1, we give an answer to the moduli problem of nondegenerate binary cubic forms with respect to (untwisted) GL2-equivalence. Theorem 0.2. Let k be a field that contains a primitive third root of unity. There is a one-to-one correspondence between { (untwisted) GL2−orbits of binary cubic forms over k }