The Elliptic Curve Decomposition of Central Conics in the Real Hyperbolic Plane

Abstract In [1] we classified the intrinsic conics in H² as distinguished subsets of the projective categories (as described in [2] and [3]) to which they belong. However, their relation within H² to these general conics was not determined beyond locus characterizations. Here we show that any central conic that is not intrinsic is the elliptic curve sum of two intrinsic central conics (properly oriented). First, we use a grid consisting of central intrinsic conics and their orthogonal trajectories to introduce coordinates for H². In any inversive model of H², these two sets of curves comprise a complete set of representatives for the genus 1 curves whose shape invariant j (Jacobi absolute invariant) is real. Given two of the intrinsic conics, their respective intersections with any orthogonal trajectory F can be paired by orientation to the axes of the conics so that the elliptic curve sum of the pair with respect to F lies on the same central conic for any F. This new conic, which is not intrinsic, is considered to be the sum of the two oriented conics. Thus, the addition of two points on F is given by an appropriate intersection with F with a central conic determined by those points, and provides an alternative construction of the group law on an elliptic curve with j≤1. Conversely, any non-intrinsic central conic decomposes as the sum of two intrinsic conics. This decomposition is unique up to split inversion, a quasi-symmetry of H² defined in [1] and reprised in Section 1.