The arithogeometric mean

Abstract

In this expository note we derive the arithogeometric mean method of Gauss. We assume the reader knows basic algebraic geometry and the theory of elliptic curves. We arrive at the required quadratic substitution with no guesswork, explain how to compute the periods of an elliptic curve, and how to compute the (inverse) elliptic integrals (elliptic functions). The algorithms described are suitable for machine computation to hundreds of digits of accuracy. Begin with an elliptic curve E given by the equation y2 = x(x + r) (x + s). We assume that (x + r) (x + s) has real coefficients, and that neither r nor s is real and nonnegative. Any real elliptic curve can be brought into this form. We let L c IE be the lattice of dx ~. periods of co = ~yy, and let ~: II;/L , E(II2) be the isomorphismoo with 4~*co = dz (z = parameter on 112). We write L = 7/. 7 + 2~6 where 7 = 2 ~ co is real. We know that 0 �9 ([0, 7]) is a component of E(N) containing ~(0) = oo, so 4~(�89 = (0, 0). The other points of order two are (- r, 0) and ( - s, 0), so we may assume 4~(�89 = ( - r, 0) and 4~(�89 + �89 = (- s, 0) (by interchanging r and s, if necessary). Let A be the lattice 7/. 7 + 7Z26 ~ IlL It is invariant under complex conjugation, so it corresponds to a real elliptic curve. We find an equation v2= u(u + R)(u + S) /x