Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



A normal form for elliptic curves

Author: Harold M. Edwards
Journal: Bull. Amer. Math. Soc. 44 (2007), 393-422
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically: April 9, 2007
MathSciNet review: 2318157
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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

$\displaystyle X = \frac 1a \cdot \frac{xy' + x'y}{1 + xyx'y'}, \quad Y = \frac 1a \cdot \frac{yy' - xx'}{1 - xyx'y'}.$

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.

References [Enhancements On Off] (What's this?)

  • 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. C. Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. 6, AMS, New York, 1951. MR 0042164 (13:64a)
  • 4. H. M. Edwards, Essays in Constructive Mathematics, Springer, New York, 2004. MR 2104015 (2005h:00010)
  • 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.
  • 7. A. Hurwitz and R. Courant, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin, 1922, 1925, 1964. MR 0173749 (30:3959)
  • 8. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York, 1990. MR 1070716 (92e:11001)
  • 9. C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti (Königsberg), 1829 (Math. Werke, vol. 1, pp. 49-241).
  • 10. A. W. Knapp, Elliptic Curves, Princeton Univ. Press, Mathematical Notes 40, 1992. MR 1193029 (93j:11032)
  • 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, Berlin, 1974. MR 0366917 (51:3163)
  • 13. J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, New York, 1992. MR 1171452 (93g:11003)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 54C40, 14E20, 46E25, 20C20

Retrieve articles in all journals with MSC (2000): 54C40, 14E20, 46E25, 20C20

Additional Information

Harold M. Edwards
Affiliation: Department of Mathematics, New York University, 251 Mercer Street, New York, New York 10012

Keywords: Elliptic curves, elliptic functions, Riemann surfaces of genus one
Received by editor(s): December 27, 2005
Published electronically: April 9, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society