Kummer Theory of Abelian Varieties and Reductions of Mordell-weil Groups

It does not appear that the torsion ambiguity can be eliminated with our present approach, and it is not clear to the author how to modify the arguments for the non-commutative case. We note that our theorem applies in particular to products of non-isogenous elliptic curves. Gajda’s question has its origins in the support problem of P. Erdős: if x and y are positive integers such that for any n ≥ 1 the set of primes dividing x−1 is the same as the set of primes dividing y−1, then must x equal y? Corrales-Rodrigáñez and Schoof gave an affirmative answer to this question in [3] and also answered the corresponding question for elliptic curves; this was generalized by Banaszak, Gajda and Krasoń in [1] to certain abelian varieties with complex or real multiplication and EndFA a commutative maximal order. In this context the support problem takes the following form.