Geodesic
On calculation of an automorphism group of a hyperelliptic curve
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 140-154 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

An algorithm for finding an automorphism group of a hyperelliptic curve $y^2 = p (x) $ with $p \in \mathbb Q [x] $ over field of complex numbers is proposed. The algorithm is based on parametric representation of the curve at singular points by the help of power series. The implementation of the algorithm in the computer algebra system Sage is presented, several examples are given. Numerical experiments have shown that the algorithm does not lead to excessively complex calculations. Used format for description of the found groups allows to apply standard for Sage instruments for the investigation of small-order groups. 
@article{ZNSL_2019_485_a7,
     author = {M. D. Malykh and L. A. Sevastianov},
     title = {On calculation of an automorphism group of a hyperelliptic curve},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {140--154},
     year = {2019},
     volume = {485},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a7/}
}
TY  - JOUR
AU  - M. D. Malykh
AU  - L. A. Sevastianov
TI  - On calculation of an automorphism group of a hyperelliptic curve
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2019
SP  - 140
EP  - 154
VL  - 485
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a7/
LA  - ru
ID  - ZNSL_2019_485_a7
ER  - 
%0 Journal Article
%A M. D. Malykh
%A L. A. Sevastianov
%T On calculation of an automorphism group of a hyperelliptic curve
%J Zapiski Nauchnykh Seminarov POMI
%D 2019
%P 140-154
%V 485
%U http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a7/
%G ru
%F ZNSL_2019_485_a7
M. D. Malykh; L. A. Sevastianov. On calculation of an automorphism group of a hyperelliptic curve. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 140-154. http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a7/

[1] R. Hartshorne, Algebraic Geometry, Springer, 1977 | MR | Zbl

[2] P. Painlevé, Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (septembre, octobre, novembre 1895) sur l'invitation de S. M. le roi de Suède et de Norwège, {ØE}uvres de Painlevé, 1, CNRS, Paris, 1971

[3] A. Broughton, T. Shaska, A. Wootton, “On automorphisms of algebraic curves”, Contemp. Math., 724, Amer. Math. Soc., Providence, RI, 2019, 175–212 | DOI | MR | Zbl

[4] Eslam E. Badr, Mohammed A. Saleem, Cyclic Automorphisms groups of genus 10 non-hyperelliptic curves, 2013, arXiv: 1307.1254

[5] Tanush Shaska, “Determining the automorphism group of a hyperelliptic curve”, Proc. of the international symposium on Symbolic and algebraic computation, 2003, 248–254 | MR | Zbl

[6] Tanush Shaska, “Some spectral families of hyperelliptic curves”, J. Algebra Its Appl., 03:1 (2004), 75–89 | DOI | MR

[7] D. Sevilla, T. Shaska, “Hyperelliptic curves with reduced automorphism group $A_5$”, Applicable Algebra in Engineering, Communication and Computing, 18:3 (2007) | DOI | MR

[8] Overview of Algcurves package, https://www.maplesoft.com/support/help/Maple/view.aspx?path=algcurves

[9] W. Stein et al., “Affine curves”, Sage Reference Manual, 2019 http://doc.sagemath.org/html/en/reference/curves/sage/schemes/curves/affine_curve.html

[10] S. A. Abramov, M. Petkovšek, “On polynomial solutions of linear partial differential and (q-)difference equations”, Computer Algebra in Scientific Computing, CASC 2012, Lecture Notes in Computer Science, 7442, 2012, 1–11 | DOI | MR | Zbl

[11] S. V. Paramonov, “Undecidability of existence of certain solutions of partial differential and difference equations”, Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, 8660, 2014, 350–356 | DOI | Zbl

[12] E. A. Ayryan, M. D. Malykh, L. A. Sevastianov, Ying Yu, “On explicit difference schemes for autonomous systems of differential equations on manifolds”, Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, 11520, 2019 | MR

[13] A. Hurwitz, “Über diejenigen algebraischen Gebilde, welche eindeutige Transformationen in sich zulassen”, Math. Ann., 32 (1887) | MR

[14] K. Weierstrass, Mathematische Werke, v. 4, Mayer, Berlin, 1902 | Zbl

[15] D. Joyner et al., “Permutation groups”, Sage Reference Manual http://doc.sagemath.org/html/en/reference/groups/sage/groups/perm_gps/permgroup.html

[16] H.G. Zeuthen, Lehrbuch der abzählenden Methoden der Geometrie, Teubner, Leipzig–Berlin, 1914 | Zbl

[17] F. Severi, Lezioni di geometria algebrica, Angelo Graghi, Padova, 1908 | MR

[18] R. Nevanlinna, Uniformisierung, Springer, Berlin, 1953 | MR | Zbl

[19] Hyperelliptic curves for Sage, https://malykhmd.neocities.org

[20] Group Names, https://people.maths.bris.ac.uk/m̃atyd/GroupNames