that all its factors are jacobian varieties. Somehow related to the above is the following converse. Assume we are given $r \geq 1$ closed Riemann surfaces $S_1, \ldots, S_r$, where $S_j$ has genus $g_j \geq 1$.

\(a) Is there a closed Riemann surface whose jacobian variety is isogenous to the product $JS_1 \times \cdots \times  JS_r \times A$, for $A$ being also a suitable product of jacobian varieties?

\(b) If the answer to (a) is affirmative, then which is minimal genus $e(S_1,\ldots,S_r)$ of such a closed Riemann surfaces? Clearly, $e(S_1,\ldots,S_r) \geq g_1 + \cdots g_r$.

In this paper, we consider the previous problems in the particular case that each $S_{j}$ is a genus one (i.e., an elliptic curve) Riemann surface $E_{j}$. In this case, we also set $e(r)$ as the maximum of the values $e(E_{1},\ldots,E_{r})$, where $E_1,\ldots,E_r$ run over the space of elliptic curves. Since $e(E_1)=e(1)=1$, the interesting case is $r \geq 2$.

In [@E-S] Ekedahl and Serre constructed examples of elliptic curves $E_1, \ldots, E_r$ (for almost every $r \leq 1297$) so that $e(E_1, \ldots, E_r)=r$. It is an open problem to determine if there are similar examples for infinitely many values of $r$ . Another similar constructions can 