Curves of Genus 2 with Split Jacobian

We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and /: X —► E a map from X to an elliptic curve, then X has split jacobian. It is not true that a complement to E in the jacobian of X is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve E' and algebraic f:X—>E', and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics 0 and 3. 0. Introduction. We say that a curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. In the later half of the nineteenth century a considerable body of work was done on the reduction of abelian integrals to elliptic. Krazer [1] gives a summary of the results obtained. Stated geometrically, the results are particular families of algebraic curves of genus 2 with maps of degree 2, 3 or 4 to elliptic curves. Such curves have split jacobian. The general question of split jacobian curves and particularly, those of genus 2, is of interest for several reasons. Split jacobian curves often have the maximal number of points over finite fields, e.g. the examples of Moret-Bailley [2] are one parameter families of curves of genus 2 over fields of order p2 whose jacobians are isomorphic to the square of the supersingular elliptic curve, and which have maximal numbers of points over fields of order p2n, n > 2. Split jacobian curves of genus 2 have also been used to exhibit nonisomorphic curves with the same jacobian; vide [3, 4]. The approach in these papers is through the algebraic geometry of abelian varieties, and the constructions are therefore far from explicit. Consider the following: Let X be a curve of genus 2, and f:X—>Ea map from X to an elliptic curve. The jacobian of X is therefore isogenous to a product of E and another elliptic curve, E1. Problem: find E', e.g. what is its j-invariant? It is not clear, nor even true (vide T. Shioda [7]), that £" is uniquely determined. However, under certain conditions there is a canonical choice of complement, and we give an algorithm for finding that curve. Our aim is to provide explicit equations for the curves and the maps between them. We obtain a fairly complete combinatorial characterization of the splitting of the jacobians of curves of genus 2. The splitting is characterized by the degree of the map / above. Jacobi, generalizing an example of Legendre, gave the complete solution for degree 2. Given any involution of P1, and three points not fixed by the involution, the curve of genus 2 which has its 6 Weierstrass points above the three points and their images under the involution, maps to the two elliptic curves represented as double covers of the quotient of P1 by the involution, ramified at Received by the editors July 20, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 14H40, 11G10. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 41 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use