Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms
نویسنده
چکیده
Let E1 and E2 be two elliptic curves over an algebraically closed field K. The purpose of this paper is to study the question of whether or not the product surface E1 × E2 can be the Jacobian of a (smooth, irreducible) curve C of genus 2. By properties of the Jacobian, this question is equivalent to the question of whether or not there is such a curve C on E1 × E2. This question was first investigated in 1965 by Hayashida and Nishi[7], [6] who obtained partial results. Later Ibukiyama, Katsura and Oort[8] settled the case that E1 and E2 are supersingular (see Theorem 5 below). In studying the moduli spaces of genus 2 curves C whose Jacobians are isomorphic to a product of two elliptic curves, the following result was obtained in [11]:
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