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Bulletin of the American Mathematical Society

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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
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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?)

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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