A normal form for elliptic curves. (English) Zbl 1134.14308

Edwards, Harold M.

Bull. Am. Math. Soc., New Ser. 44, No. 3, 393-422 (2007).

Summary: The normal form \(x^2 + y^2 = a^2 + a^2x^2y^2\) for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be stated explicitly
\[ X = \frac 1a \cdot \frac{xy' + x'y}{1 + xyx'y'}, \quad Y = \frac 1a \cdot \frac{yy' - xx'}{1 - xyx'y'}. \]
The \( j\)-invariant of an elliptic curve determines \(24\) values of \(a\) for which the curve is equivalent to \( x^2 + y^2 = a^2 + a^2x^2y^2\), namely, the roots of \( (x^8 + 14x^4 \) \( + 1)^3 - \frac j{16}(x^5 - x)^4\). The symmetry in \( x\) and \( y\) implies that the two transcendental functions \( x(t)\) and \( y(t)\) that parameterize \( x^2 + y^2 = a^2 + a^2x^2y^2\) in a natural way are essentially the same function, just as the parameterizing functions \( \sin t\) and \( \cos t\) of the circle are essentially the same function. Such a parameterizing function is given explicitly by a quotient of two simple theta series depending on a parameter \( \tau\) in the upper half plane.

MSC:

14H52	Elliptic curves
33E05	Elliptic functions and integrals

Keywords:

elliptic curves; elliptic functions; Riemann surfaces of genus one

References:

[1]	N. H. Abel, Recherches sur les fonctions elliptiques, Crelle, vols. 2, 3, Berlin, 1827, 1828; Oeuvres, I, pp. 263-388.
[2]	N. H. Abel, Mémoire sur une propriété générale d’une classe très-étendue de fonctions transcendantes, Mémoires présenteés par divers savants à l’Académie des sciences, Paris, 1841. Also Oeuvres Complètes, vol. 1, 145-211.
[3]	Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. · Zbl 0045.32301
[4]	Harold M. Edwards, Essays in constructive mathematics, Springer-Verlag, New York, 2005. · Zbl 1090.11001
[5]	L. Euler, Observationes de Comparatione Arcuum Curvarum Irrectificabilium, Novi Comm. Acad. Sci. Petropolitanae, vol. 6, pp. 58-84, 1761, Opera, ser. 1, vol. 20, pp. 80-107, Eneström listing 252.
[6]	C. F. Gauss, Werke, Vol. 3, p. 404. · JFM 50.0001.01
[7]	Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl. Vierte vermehrte und verbesserte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). · JFM 51.0236.12
[8]	Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. · Zbl 0712.11001
[9]	C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti (Königsberg), 1829 (Math. Werke, vol. 1, pp. 49-241).
[10]	Anthony W. Knapp, Elliptic curves, Mathematical Notes, vol. 40, Princeton University Press, Princeton, NJ, 1992. · Zbl 0804.14013
[11]	B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. der Berliner Akad., Nov. 1859; Werke, 145-153.
[12]	I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. · Zbl 0284.14001
[13]	Joseph H. Silverman and John Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. · Zbl 0752.14034