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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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A normal form for elliptic curves
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by Harold M. Edwards PDF
Bull. Amer. Math. Soc. 44 (2007), 393-422 Request permission


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.
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Additional Information
  • Harold M. Edwards
  • Affiliation: Department of Mathematics, New York University, 251 Mercer Street, New York, New York 10012
  • Received by editor(s): December 27, 2005
  • Published electronically: April 9, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 44 (2007), 393-422
  • MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
  • DOI:
  • MathSciNet review: 2318157