Torsion points and matrices defining elliptic curves

Abstract

Let k be an algebraically closed field, char k ≠ 2, 3, and let X ⊂ ℙ2 be an elliptic curve with defining polynomial f. We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r × 3r with linear polynomial entries such that det Φr = fr. We also show that the identity, thought of as the trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Ψr of size (3r - 2) × (3r - 2) with linear and quadratic polynomial entries such that det Ψr = fr.

Indecomposable matrices defining plane cubics

“…The main result of this paper is Theorem 3.2 containing the construction of the three indecomposable pfaffian representations for a Weierstrass cubic. We outline similar constructions for indecomposable determinantal representations of order r ≥ 2 corresponding to indecomposable vector bundles of rank r. These computations are an appendix to the paper of Ravindra and Tripathi [16]. As a corollary we verify the Kippenhahn conjecture for M 6 .…”

“…The three non-block matrices in Theorem 3.2 are by the above the only indecomposable skew-symmetric matrices with determinant F 2 . For r = 3 Ravindra and Tripathi [16] proved the existence of eight indecomposable 9 × 9 determinantal representations of C. The corresponding indecomposable cokernels are extensions of the nontrivial 3−torsion points on JC with themselves (i.e., the eight flexes on the affine Weierstrass cubic). In order to explicitly construct these determinantal representations, we would need to repeat the proof of Theorem 3.2 for 9 × 9 matrices.…”

“…In particular, on a given cubic curve the number of indecomposable ACM bundles of rank r with trivial determinant equals to the number of r-torsion points. Recently, Ravindra and Tripathi [16] used these indecomposable vector bundles to prove the existence of indecomposable determinantal representations of order greater than two.…”

Remarks on higher-rank ACM bundles on hypersurfaces

“…A smooth plane cubic is an elliptic curve; its torsion points of odd order r, give rise to Ulrich bundles of rank r, (cf. [15]). Similarly, lines in quadric surfaces in P 3 correspond to ACM line bundles not isomorphic to O X (c) for any c ∈ Z.…”