Modular invariant and good reduction of elliptic curves (Q810603)

Let K be a number field and let S be a finite set of places of K. The authors obtain an explicit description of the set \(J_ S\) of the elements j in K such that there exists an elliptic curve E defined over K, with modular invariant j(E) equal to j and good reduction at all places outside S. This is done by computing first, for any place v of K, the set: \[ J(v) = \{j\in K:\;\exists E\text{ with } j(E)=j \text{ and good reduction at} v\}, \] and by computing then the local-to-global obstruction for an element in \(\cup_{v\not\in S}J(v) \) to belong to \(J_ S\).