David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of a hyperelliptic curve?
How is this related to David's question? Well, if we can multiply two elliptic curves over $\mathbb{Q}(t)$ with large rank, and the result is isogenous to the jacobian of a hyperelliptic curve, then this will probably produce record families answering David's question, i.e. genus two curves with very large rank. It is also interesting for all genera, so don't restrict answers to 2. On the other hand, answers containing arithmetic information, for example on elliptic curves over the rationals, are more than welcome.
Best Answer
Over the complex numbers every product of two elliptic curves is isogenous to the jacobian of a genus 2 (hyperelliptic) curve. Indeed, the corresponding Siegel upper half-space $H_2$ is an orbit of the real symplectic group $Sp(4,R)$. Since the subgroup $Sp(4,Q)$ of its rational points is everywhere dense in $Sp(4,R)$, every $Sp(4,Q)$-orbit is everywhere dense in $H_2$ and therefore meets the (Torelli) open subset $T_2$ of $H_2$ that ``parametrized" the Jacobians. Now one has only to notice that if points $x,y \in H_2$ correspond to principally polarized abelian surfaces $A_x$ and $A_y$ respectively then $A_y$ is isogenous to $A_x$ if $y$ lies in the $Sp(4,Q)$-orbit of $x$. (And, of course, one should take $x$ with $A_x$ being a product of two elliptic curves.)
Similar arguments prove that every product of three elliptic curves is isogenous to the jacobian of a genus 3 curve. (These arguments were used in Sect. 2, Remark 3 on pp. 60--61 of arXiv:0912.4325v1 [math.NT] in order to prove certain properties of the modular height.)