Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms
The author addresses the problem raised in [\textit{T. Hayshida} and \textit{M. Nishi}, J. Math. Soc. Japan 17, 1--16 (1965; Zbl 0132.41701)]: Whether or not the product \(E_1\times E_2\) of two elliptic curves over an algebraically closed field can be the Jacobian of a smooth irreducible curve \(C\) of genus 2 (this is equivalent to the question of whether or not there is such a curve \(C\) on \(E_1\times E_2\)). The author translates this problem into a classification problem about quadratic and ternary forms and answers the above question in the case that \(E_1\) and \(E_2\) are isogenous elliptic curves with complex multiplication. Namely, let \(q_{E_1, E_2}\) be the quadratic form on \(\text{Hom}(E_1, E_2)\) defined by \(q_{E_1, E_2}(f):=\text{deg}(f)\). It is proved in the paper under review that there is no genus 2 curve on \(E_1\times E_1\) if and only if \(q_{E_1, E_2}\) is equivalent to one of the 15 forms \(ax^2+bxy+cy^2\) whose coefficients \((a, b, c)\) are in the following list: \[ \{k(1,1,1): k=1,2,4,6,10\}\cup\{k(1,0,1) : k=1,2, 6\}\cup \{(1, 1, 2), (1, 1, 4)\} \] \[ \cup\{2(1,1,c): c=3, 9\}\cup \{2(1,0,c): c=2,5\}\cup \{2(2,0,3)\}. \]
elliptic curves with complex multiplication
scientific article; zbMATH DE number 6308466
scientific article