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

Abstract

Let $F/k$ be a cyclic extension of number fields of prime degree. Let $\rho$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $\pi$ a cuspidal automorphic representation of GL${}_2({\mathbb{A}}_F)$, such that the $L$-functions $L(s,{\rho }_v)$ and $L(s,{\pi }_v)$ agree at all (but finitely many of) the places $v$ of degree one over $k$. We prove in this case that we have the global identity $L(s,\rho )=L(s,\pi )$, with ${\rho }_v\leftrightarrow {\pi }_v$ being given by the local Langlands correspondence at all $v$. In particular, $\pi$ is tempered and $L(s,\rho )$ is entire.

arXiv comment12 pages, to appear in Contemp. Math. volume in honor of Jim Cogdell's 60th birthday