A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes

Let F be a number field and ρ an irreducible Galois representation of Artin type, i.e., ρ is a continuous C-representation of the absolute Galois group ΓF . Suppose π is a cuspidal automorphic representation of GLn(AF ) such that the L-functions L(s, ρ) and L(s, π) agree outside a set S of primes. (Here, these L-functions denote Euler products over just the finite primes, so that we may view them as Dirichlet series in a right half plane.) When S is finite, the argument in Theorem 4.6 of [DS74] implies these two L-functions in fact agree at all places (cf. Appendix A of [Mar04]). We investigate what happens when S is infinite of relative degree ≥ 2, hence density 0, for the test case n = 2. Needless to say, if we already knew how to attach a Galois representation ρ′ to π with L(s, ρ′) = L(s, π) (up to a finite number of Euler factors), as is the case when F is totally real and π is generated by a Hilbert modular form of weight one ([Wil88], [RT83]), the desired result would follow immediately from Tchebotarev’s theorem, as the Frobenius classes at degree one primes generate the Galois group. Equally, if we knew that ρ is modular attached to a cusp form π′, whose existence is known for F = Q and ρ odd by Khare–Wintenberger ([Kha10]), then one can compare π and π′ using [Ram94]. However, the situation is more complex if F is not totally real or (even for F totally real) if ρ is even. Hopefully, this points to a potential utility of our approach. We prove the following