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

摘要

Introduction Let F be a number field and p an irreducible Galois representation of Artin type,i.e.,p is a continuous C-representation of the absolute Galois group Γ_F. Suppose π is a cuspidal automorphic representation of GL_n (A_F) such that the L-functions L (s,p) 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 p' to 7r with L (s,p') =L (s,7r) (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 p is modular attached to a cusp form 7r',whose existence is known for F= Q and p 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 p is even. Hopefully,this points to a potential utility of our approach.