Jacobians isomorphic to a product of two elliptic curves

In 1965 Hayashida and Nishi initiated the study of genus 2 curves C whose Jacobian JC is isomorphic to a product A = E1×E2 of two elliptic curves. In their papers [12], [14] and [13], they determined the number of curves C with JC ' A for a fixed A in many cases, thereby exhibiting the existence of such curves. A similar count was done for supersingular curves by Ibukiyama, Katsura and Oort[16]. Recently there has been renewed interest in such curves, particularly in connection with moduli problems; cf. Earle[6], Lange[26], and McMullen[28], [29]. The purpose of this article is determine how such curves are distributed in the moduli space M2 of genus 2 curves over an algebraically closed field K. By a result of Lange[25] it is known that these lie on infinitely many curves in M2; see also [6]. Here we want to make the nature of these curves precise. To this end, let us say that a curve C has type d if JC ' E1 × E2, where E1 and E2 are connected by a cyclic isogeny of degree d. (If E1 has CM or is supersingular, then this definition has to slightly modified; see §4 below.) Since every curve C with JC ' E1×E2 has some type d ≥ 1 (cf. Proposition 25), the following result describes the set of all such curves in M2: