Geodesic
Multiplication and division on elliptic curves, torsion points and roots of modular equations
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 24-57
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Having expressed the ratio of the length of the Lemniscate of Bernoulli to the length of its cocentred superscribing circle as the reciprocal of the arithmetic-geometric mean of $1$ and $\sqrt{2}$, Gauss wrote in his diary, on May 30, 1799, that thereby “an entirely new field of analysis” emerges. Yet, up to these days, the study of elliptic functions (and curves) has been based on two traditional approaches (namely, that of Jacobi and that of Weiestrass), rather than a single unifying approach. Replacing artificial dichotomy by a, methodologically justified, single unifying approach does not only enable re-deriving classical results eloquently but it allows for undertaking new calculations, which did seem either unfeasible or too cumbersome to be explicitly performed. Here, we shall derive readily verifiable explicit formulas for carrying out highly efficient arithmetic on complex projective elliptic curves. We shall explicitly relate calculating the roots of the modular equation of level $p$ to calculating the $p$-torsin points on a corresponding elliptic curve, and we shall re-bring to light Galois exceptional, never nearly surpassable and far from fully appreciated, impact. 
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S. Adlaj. Multiplication and division on elliptic curves, torsion points and roots of modular equations. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 24-57. http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a1/

[1] S. Adlaj, “Galois elliptic function and its symmetries”, $12^{\rm th}$ International Conference on Polynomial Computer Algebra, ed. Vassiliev N.N., St. Petersburg department of Steklov Institute of Mathematics, St. Petersburg, 2019, 11–17

[2] S. Adlaj, “On the second memoir of Évariste Galois' last letter”, Computer Tools in Science and Education, 4 (2018), 5–20 | DOI

[3] S. Adlaj, Thread equilibrium in a linear parallel force field, LAP LAMBERT, Saarbrucken, 2018 (in Russian)

[4] S. Adlaj, “An analytic unifying formula of oscillatory and rotary motion of a simple pendulum (dedicated to $70^{\rm th}$ birthday of Jan Jerzy Slawianowski”, Proc. Int. Conf. “Geometry, Integrability, Mechanics and Quantization”, eds. Mladenov I., Ludu A., Yoshioka A., Avangard Prima, Sofia, Bulgaria, 2015, 160–171 | MR

[5] S. Adlaj, Torsion points on elliptic curves and modular polynomial symmetries, Presented at the joined MSU-CCRAS Computer Algebra Seminar (September 24, 2014) http://www.ccas.ru/sabramov/seminar/lib/exe/fetch.php?media=adlaj140924.pdf

[6] S. Adlaj, “Modular Polynomial Symmetries”, $7^{\rm th}$ International Conference on Polynomial Computer Algebra, ed. Vassiliev N.N., St. Petersburg department of Steklov Institute of Mathematics, St. Petersburg, 2014, 4–5

[7] S. Adlaj, “Elliptic and coelliptic polynomials”, $17^{\rm th}$ Workshop on Computer Algebra (Dubna, Russia, May 21-22, 2014) http://compalg.jinr.ru/Dubna2014/abstracts2014_files/1159.pdf

[8] S. Adlaj, “Mechanical interpretation of negative and imaginary tension of a tether in a linear parallel force field”, Selected Works of the International Scientific Conference on Mechanics “Sixth Polyakhov Readings”, Saint-Petersburg State University, Saint-Petersburg, 2012, 13–18

[9] S. Adlaj, “Iterative algorithm for computing an elliptic integral”, Issues on motion stability and stabilization, 2011, 104–110 (in Russian)

[10] S. Adlaj, Eighth lattice points, arXiv: 1110.1743

[11] E. Betti, “Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado primo”, Dagli Annali di Scienze matimatiche e fisiche (Roma, 1851), v. II, 5–19

[12] E. Betti, “Un teorema sulla risoluzione analitica delle equazioni algebriche”, Dagli Annali di Scienze matimatiche e fisiche (Roma, 1854), v. V, 10–17

[13] D. Cox, “The arithmetic-geometric mean of Gauss”, Mathématique, 30 (1984), 275–330 | MR | Zbl

[14] É. Galois, “Lettre de Galois G M. Auguste Chevalier”, Journal de Mathématiques Pures et Appliquées, XI (1846), 408–415

[15] C. Hermite, “Sur la résolution de l'équation du cinquième degré”, Comptes Rendus de l'Académie des Sciences, XLVI:I (1858), 508–515

[16] S. D. Meshveliani, “Computer proof for a “mysterious” equality for quartic roots”, $17^{\rm th}$ Workshop on Computer Algebra (Dubna, Russia, May 21-22, 2014) http://compalg.jinr.ru/Dubna2014/abstracts2014_files/1199.pdf

[17] J-P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1973 | MR | Zbl

[18] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, 1981 | MR

[19] L. A. Sohnke, “Aequationes modulares pro transformatione Functionum Ellipticarum”, Journal für die reine und angewandte Mathematik, 16 (1837), 97–130 | MR