publication . Article . Preprint . 2019

Numerical computation of endomorphism rings of Jacobians

Bruin, Nils; Sijsling, Jeroen; Zotine, Alexandre;
Open Access
  • Published: 28 Jan 2019 Journal: The Open Book Series, volume 2, pages 155-171 (issn: 2329-9061, eissn: 2329-907X, Copyright policy)
  • Publisher: Mathematical Sciences Publishers
Abstract
We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.
Subjects
arXiv: Mathematics::Algebraic Geometry
free text keywords: Mathematics - Number Theory, Mathematics - Algebraic Geometry, 14H40, 14H37, 14H55, 14Q05
Funded by
NSERC
Project
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
19 references, page 1 of 2

[1] E. Arbarello, M. Cornalba, P. A. Gri ths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985.

[2] Franz Aurenhammer, Voronoi diagrams{a survey of a fundamental geometric data structure, ACM Comput. Surv. 23 (September 1991), no. 3, 345{405. [OpenAIRE]

[3] David H. Bailey, Karthik Jeyabalan, and Xiaoye S. Li, A comparison of three high-precision quadrature schemes, Experiment. Math. 14 (2005), no. 3, 317{329.

[4] H. F. Baker, Examples of the application of Newton's polygon to the theory of singular points of algebraic functions, Transactions of the Cambridge Philosophical Society 15 (1893), 403.

[5] Christina Birkenhake and Herbert Lange, Complex abelian varieties, Second, Grundlehren der Mathematischen Wissenschaften, vol. 302, Springer-Verlag, Berlin, 2004.

[6] I. Bogaert, Iteration-free computation of Gauss-Legendre quadrature nodes and weights, SIAM J. Sci. Comput. 36 (2014), no. 3, A1008{A1026.

[7] Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235{254.

[8] Nils Bruin and Emre Sertoz, Prym varieties of genus 4 curves, 2018. In preparation.

[9] Nils Bruin, Jeroen Sijsling, and Alexandre Zotine, Calculations with numerical Jacobians, 2018. https://github.com/nbruin/examplesNumericalEndomorphisms.

[10] Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight, Rigorous computation of the endomorphism ring of a Jacobian, 2016. arXiv:1707.01158.

[11] Wolfram Decker, Gert-Martin Greuel, Gerhard P ster, and Hans Schonemann, Singular 4-1-1 | A computer algebra system for polynomial computations, 2018.

[12] Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28{46. Advances in nonlinear mathematics and science.

[13] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), no. 170, 463{471. [OpenAIRE]

[14] G. Frobenius, Theorie der linearen Formen mit ganzen Coe cienten, Crelle 86 (1879), 146{208.

[15] F. Hess, An algorithm for computing isomorphisms of algebraic function elds, Algorithmic number theory, 2004, pp. 263{271.

19 references, page 1 of 2
Abstract
We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.
Subjects
arXiv: Mathematics::Algebraic Geometry
free text keywords: Mathematics - Number Theory, Mathematics - Algebraic Geometry, 14H40, 14H37, 14H55, 14Q05
Funded by
NSERC
Project
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
19 references, page 1 of 2

[1] E. Arbarello, M. Cornalba, P. A. Gri ths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985.

[2] Franz Aurenhammer, Voronoi diagrams{a survey of a fundamental geometric data structure, ACM Comput. Surv. 23 (September 1991), no. 3, 345{405. [OpenAIRE]

[3] David H. Bailey, Karthik Jeyabalan, and Xiaoye S. Li, A comparison of three high-precision quadrature schemes, Experiment. Math. 14 (2005), no. 3, 317{329.

[4] H. F. Baker, Examples of the application of Newton's polygon to the theory of singular points of algebraic functions, Transactions of the Cambridge Philosophical Society 15 (1893), 403.

[5] Christina Birkenhake and Herbert Lange, Complex abelian varieties, Second, Grundlehren der Mathematischen Wissenschaften, vol. 302, Springer-Verlag, Berlin, 2004.

[6] I. Bogaert, Iteration-free computation of Gauss-Legendre quadrature nodes and weights, SIAM J. Sci. Comput. 36 (2014), no. 3, A1008{A1026.

[7] Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235{254.

[8] Nils Bruin and Emre Sertoz, Prym varieties of genus 4 curves, 2018. In preparation.

[9] Nils Bruin, Jeroen Sijsling, and Alexandre Zotine, Calculations with numerical Jacobians, 2018. https://github.com/nbruin/examplesNumericalEndomorphisms.

[10] Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight, Rigorous computation of the endomorphism ring of a Jacobian, 2016. arXiv:1707.01158.

[11] Wolfram Decker, Gert-Martin Greuel, Gerhard P ster, and Hans Schonemann, Singular 4-1-1 | A computer algebra system for polynomial computations, 2018.

[12] Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28{46. Advances in nonlinear mathematics and science.

[13] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), no. 170, 463{471. [OpenAIRE]

[14] G. Frobenius, Theorie der linearen Formen mit ganzen Coe cienten, Crelle 86 (1879), 146{208.

[15] F. Hess, An algorithm for computing isomorphisms of algebraic function elds, Algorithmic number theory, 2004, pp. 263{271.

19 references, page 1 of 2
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue