Picard-Fuchs equations for elliptic modular varieties (Q1180325)

Let \(E_ 0\) be the union of the regular fibers of an elliptic modular surface \(\pi:E\to X\) over a compact Riemann surface \(X\), and let \(X_ 0=\pi(E_ 0)\). Then, for each positive integer \(n\), an elliptic modular variety \(E^ n\) is obtained by resolving the singularities of the compactification of the \(n\)-fold fiber product of \(E_ 0\) over \(X_ 0\). The morphism \(\pi\) induces the morphism \(\pi^ n:E^ n\to X\) and the generic fiber of \(\pi^ n\) is the product of \(n\) elliptic curves. In this paper, the Picard-Fuchs equations for the elliptic modular varieties for \(E^ 3\) and \(E^ 4\) (or, more precisely, for the fibrations \(\pi^ 3\) and \(\pi^ 4)\) are determined in terms of the Picard-Fuchs equation of the elliptic fibration \(\pi:E\to X\).