A normal form for elliptic curves

Abstract

The normal form x 2 + y 2 = a 2 + a 2 x 2 y 2 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 \[ X = 1 a ⋅ x y ′ + x ′ y 1 + x y x ′ y ′ , Y = 1 a ⋅ y y ′ − x x ′ 1 − x y x ′ y ′ . X = \frac 1a \cdot \frac {xy’ + x’y}{1 + xyx’y’}, \quad Y = \frac 1a \cdot \frac {yy’ - xx’}{1 - xyx’y’}. \] The j j -invariant of an elliptic curve determines 24 values of a a for which the curve is equivalent to x 2 + y 2 = a 2 + a 2 x 2 y 2 x^2 + y^2 = a^2 + a^2x^2y^2 , namely, the roots of ( x 8 + 14 x 4 (x^8 + 14x^4 + 1 ) 3 − j 16 ( x 5 − x ) 4 + 1)^3 - \frac j{16}(x^5 - x)^4 . The symmetry in x x and y y implies that the two transcendental functions x ( t ) x(t) and y ( t ) y(t) that parameterize x 2 + y 2 = a 2 + a 2 x 2 y 2 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 \sin t and cos ⁡ t \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 \tau in the upper half plane.