Title:
an effective open image theorem for elliptic curves over the rationals
without CM
Abstract:
A crucial hypothesis to apply Kolyvagin's theorem is the surjectivity of
the mod-p Galois representation of E/Q without CM. We prove an explicit
bound p_0(N) only depending on the conductor N of E such that the mod-p
Galois representation is surjective if p > p_0(N). The proof is based on
an argument due to Serre, which uses Mazur's classification of rational
isogenies of prime degree, the density of supersingular primes for E/Q
and the modularity of elliptic curves over the rationals.