A normal form for elliptic curves

Abstract

The normal form x2 + y2 = a2+a 2x2y2 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 = 1/a · xy′+x′y/1+xyx′y′, Y = 1/a · yy′-xx′/1-xyx′y′. The j-invariant of an elliptic curve determines 24 values of a for which the curve is equivalent to x2 + y2 = a2 + a2x2y 2, namely, the roots of (x8 + 14x4 + 1) 3 - j/16(x5 - x)4. The symmetry in x and y implies that the two transcendental functions x(t) and y(t) that parameterize x2 + y2 = a2 + a2x2y 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 τ in the upper half plane. © 2007 American Mathematical Society.