Pairs of elliptic curves with maximal Galois representations

Using a multi-dimensional large sieve inequality, we prove that, for almost all pairs (or indeed almost all k-tuples) of elliptic curves, the associated Galois representation on their torsion has maximal image. This generalizes the authorʼs previous work and provides evidence for an affirmative answer to a higher-dimensional analogue of Serreʼs uniformity question for single elliptic curves. Furthermore, as a consequence of our main theorem, one deduces the triviality of the Brauer group of the Kummer surface attached to almost all pairs of elliptic curves.