Ramified primes in the field of definition for the Mordell-Weil group of an elliptic surface

Abstract:

Let $\pi :X \to C$ be an elliptic surface defined over a number field $k$. We consider the field $K$ in which all the sections are defined. Assuming that the $j$-invariant is nonconstant, $K$ is again a number field. We describe the primes of possible ramification of the extension $K/k$ in terms of the configuration of the points of bad fibers in $C$. Aside from few possible exceptions, $K/k$ is unramified outside of the primes of bad reduction of $C$ and the primes $\mathfrak {p}$ for which two or more points of bad fibers become identical modulo $\mathfrak {p}$.