A geometric approach to Euler's addition formula : A direct formulation of the addition law for elliptic curves given in terms of Weierstra quartics

Plane quartic curves given by equations of the form y 2=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form ∫ (1/√P(x))dx with a quartic polynomial P can be derived directly from this addition law. © 2011 Springer-Verlag.