show/hide this revision's text 3 added 10 characters in body

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, partially answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.
show/hide this revision's text 2 added 5 characters in body

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?curve?"

The results result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.
show/hide this revision's text 1

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17 answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?

The results is the following:

THEOREM Consider a product of elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.