Unramified sections of the Legendre scheme and modular forms (Q2032813)

Let \(\lambda\) be a complex number \(\not\in\{0,1\}\), and consider the Legendre elliptic scheme given by its affine model \(y^2=x(x-1)(x-\lambda)\). Each torsion section of unrestricted order is defined by algebraic functions of \(\lambda\) unramified outside \(\{0,1,\infty\}\), and by Shioda's result, it is shown that every algebraic section unramified outside \(\{0,1,\infty\}\) is torsion. As one of the main results of the paper under review, the authors present an alternative proof of Shioda's result, in terms of unramified coverings of \(\mathbb P_1 - \{0,1,\infty\}\), associated to finite index subgroups of its monodromy group. Another main result of the paper can be formulated as follows where \(\pi_1\) denotes a fundamental group: Let \(\mathcal E \to B\) be a non-isotrivial elliptic scheme where \(B\) is a smooth affine complex curve. If a rational section \(\sigma : B \to \mathcal E\) is non-torsion, then the associated homomorphism: \(\pi_1(B,b_0) \to \pi_1(\mathcal E,\sigma(b_0))\) does not factor through the mapping class group. Stated above is a topological version of their theorem, and they introduce different versions as well, one of which presents information on the monodromy of the Picard-Fuchs differential equation satisfied by the periods and the logarithm of the section. The authors also point out that the theorem has applications to arbitrary non-isotrivial elliptic schemes.