The author considers cubic polynomials \(f(z)=z^2+az+b\) defined over \(\mathbb{C}(\lambda)\), with a marked point \(z_1\) of period \(N\) and multiplier \(\lambda\). The result obtained is that, for \(N=1\), there are infinitely many of such objects, and in the case \(N=3\), only finitely many of them. The case \(N=2\) has a particulary rich structure; for example, the author describes such cubic polynomials over the field given as the union of \(\mathbb{C}(\lambda^{1/n})\) for all \(n\). To study the set of cubic polynomials with marked point of period \(N\), the author introduces the 2-dimensional (one-dimensional over \(\mathbb{C}(\lambda)\)) muduli space \(\mathcal{P}_3(N)\) and considers the multiplier fibration \[ \lambda : \mathcal{P}_3(N) \rightarrow \widehat{\mathbb{C}}, \] taking a cycle to its multiplier. The question translates then into studying the existence and structure of holomorphic multiplier sections and/or probably some meromorphic sections of interest. The version of the moduli space of polynomials presented is particularly adaptable to normal forms of polynomials.
The author introduces some of the geometric properties of the modular curves \(X_1(N)\) and \(X_0(N)\) over \(\mathbb{C}(\lambda)\). In the particular case of \(N=2\), the geometry of the modular curves is related to the geometry of two non-isotrivial elliptic curves, providing points in \(X_1(2)\) and \(X_0(2)\) with an additional structure of finitely generated abelian group. The precise group structure of the Mordell-Weil group is explicitly described in the paper for the root case \(\lambda=w^n\). To do the precise estimate, the author improves a result of Fastenberg on computing the Mordel-Weil rank of cyclic covers of elliptic surfaces.