Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p is an element of S, let Q(S)((p)) be the compositum of all extensions unramified outside S of the form Q(mu(p), p root d), for d is an element of Q(x). If (sigma) = (sigma(1),..., sigma(n)) is an element of Gal((Q) over bar /Q)(n), let (Q(S)((p)))((sigma)) be the intersection of the fixed fields by <sigma(i)>, for i = 1,..., n. We provide a wide family of elliptic curves E/Q such that the rank of E((Q(S)((p)))((sigma))) is infinite for all n >= 0 and all (sigma) is an element of Gal((Q) over bar /Q)(n), subject to the parity conjecture. Similarly, let (A/Q, phi) be a polarized abelian variety, let K be a quadratic number field fixed by (sigma) is an element of Gal((Q) over bar /Q)(n), let S be an infinite set of primes of Q and let K-S(p-dihe) be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over Q. We show that, under certain hypotheses, the Z(p)-corank of Sel(p infinity) (A/F) is unbounded over finite extensions F/K contained in (K-S(p-dihe))((sigma))/K. As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.