A Rational Map between Two Threefolds

In Remark 4.5 of [4] it is mentioned that there is a piece of the cohomology H(E) which has the same L-series as that of H3(Ṽ33). According to Tate’s conjecture there should be a correspondence between V33 and E . We give such a correspondence explicitly. Other relatives of these varieties are the fibered squares of the universal curves for Γ(3) and Γ0(9). In [8] and in the section 13 of [9] Schoen constructs correspondences between these varieties and E. We refer the reader to Remark 4.5 of [4] for a list of threefolds which have a two dimensional Galois representation in H whose L-series is associated to an elliptic modular form of weight 4, and known correspondences among those varieties. We include the relevant works in the reference.