Abelian varieties over number fields, tame ramification and big Galois image
Sara Arias‐de‐Reyna, Christian Kappen · Mathematical Research Letters · 2013
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 representationis 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.