The moduli spaces of 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 [15], [17] and [16], 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[19]. Recently there has been renewed interest in such curves, particularly in connection with moduli problems; cf. Earle[7], Lange[30], and McMullen[32], [33]. 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[29] it is known that these lie on countably many curves in M2; see also [7]. 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 26), the following result describes the set of all such curves in M2: Theorem 1 The set T (d) ⊂ M2 of curves of type d is a closed subset of M2. If T (d) is non-empty, then T (d) is a finite union of irreducible curves. Moreover, if char(K) d, then each such component is birationally isomorphic either to the modular curve X0(d) + or to a degree 2 quotient thereof.