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An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields. (English) Zbl 1409.14097
Summary: We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers.

MSC:
14Q05 Computational aspects of algebraic curves
11G05 Elliptic curves over global fields
14M12 Determinantal varieties
14H50 Plane and space curves
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
Software:
RelBrauer
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References:
[1] V. Vinnikov, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl., 125, 103-140, (1989) · Zbl 0704.14041
[2] I. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012. · Zbl 1252.14001
[3] L. Galinat, Orlov’s equivalence and maximal Cohen-Macaulay modules over the mone of an elliptic curve, Math. Nachr., 287, 1438-1455, (2014) · Zbl 1329.14034
[4] A. Deajim; D. Grant, Space time codes and non-associative division algebras arising from elliptic curves, Contemp. Math., 463, 29-44, (2008) · Zbl 1163.17005
[5] T. Fisher; R. Newton, Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve, Int. J. Number Theory, 10, 1881-1907, (2014) · Zbl 1314.11042
[6] Y. Ishitsuka and T. Ito, The local-global principle for symmetric determinantal representations of smooth plane curves, Published online in The Ramanujan J., 2016. · Zbl 1368.14046
[7] Y. Ishitsuka, A positive proportion of cubic curves over \(\mathbb{Q}\) admit linear determinantal representations, arXiv:1512.05167 [math.NT], 2015.
[8] A. Beauville, Determinantal hypersurfaces, Dedicated to W. Fulton on the occasion of his 60th birthday, Michigan Math. J., 48, 39-64, (2000)
[9] M. Ciperiani; D. Krashen, Relative Brauer groups of genus 1 curves, Israel J. Math., 192, 921-949, (2012) · Zbl 1259.14020
[10] Y. Ishitsuka, Linear determinantal representations of smooth plane cubics over finite fields, arXiv:1604.00115 [math.GA], 2016.
[11] S. Lang, Abelian varieties over finite fields, Proc. Natl. Acad. Sci. USA, 41, 174-176, (1955) · Zbl 0064.03902
[12] R. Schoof, Nonsingular plane cubic curves over finite fields, J. Comb. Theory A, 46, 183-211, (1987) · Zbl 0632.14021
[13] M. Artin; F. Rodriguez-Villegas; J. Tate, On the Jacobians of plane cubics, Adv. Math., 198, 366-382, (2005) · Zbl 1092.14054
[14] Y. Ishitsuka; T. Ito, On the symmetric determinantal representations of the Fermat curves of prime degree, Int. J. Number Theory, 12, 955-967, (2016) · Zbl 1415.11062
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