Artin's conjecture for Abelian varieties with Frobenius condition

Let A be an abelian variety over a number field K of dimension r , a 1 , … , a g ∈ A ( K ) and F / K a finite Galois extension. We consider the density of primes p of K such that the quotient A ¯ ( k ( p ) ) / 〈 a ¯ 1 , … , a ¯ g 〉 has at most 2 r − 1 cyclic components and p satisfies a Frobenius condition with respect to F / K , where A ¯ is the reduction of A modulo p , k ( p ) is the residue field of p and 〈 a ¯ 1 , … , a ¯ g 〉 is the subgroup generated by the reductions a ¯ 1 , … , a ¯ g . We develop a general framework to prove the existence of this density under the Generalized Riemann Hypothesis.