On the rank of Mordell–Weil groups of abelian varieties of GL(2)-type, preprint, 2014 posted at http://www.math.ucla.edu/~hida/ConstWM.pdf

Abstract. Pick an elliptic curve E of conductor N defined over Q with good ordinary reduction at a prime p. We suppose that E is not anomalous at p up to quadratic unramified twists. Suppose that E(k) is finite for a number field k and p is outside a finite explicit set of primes (independent of k). We will prove that almost all Q-simple abelian varieties A of GL(2)-type (with prime-to-p conductor N) has finite A(k), as long as the p-divisible group A[p∞] contains a Galois module isomorphic to E[p](Q). We also give a positive rank version of this result. 1