ABELIAN VARIETIES OVER NUMBER FIELDS, TAME RAMIFICATION AND BIG GALOIS IMAGE

Given a natural number n >= 1 and a number field K, we show the existence of an integer l(0) such that for any prime number l >= l(0), there exists a finite extension F/K, unramified in all places above l, together with a principally polarized abelian variety A of dimension n over F such that the resulting l- torsion representation rho(A,l) : G(F) -> GSp(A[l]((F) over bar)) is surjective and everywhere tamely ramified. In particular, we realize GSp(2n)(F-l) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above l.