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

TL;DR

This paper proves that for a cyclic extension of number fields, if automorphic and Artin L-functions agree at almost all degree one primes, then they are equal globally, confirming a case of the Langlands correspondence.

Contribution

It establishes the global equality of L-functions for GL(2)-type automorphic and Artin representations under local agreement at degree one primes in cyclic extensions.

Findings

L-functions agree globally under local agreement at degree one primes

Automorphic representation is tempered

L(s,ρ) is entire

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.