# Base-Change and Automorphic-Induction for $GL_1$

Dear all, I try to understand the base-change and automorphic-induction in the theory of automorphic forms, for the simplest case: $GL_1$. Both are implied by Langlands conjectures

Base-Change

Let $L/K$ be an abelian extension of number fields of degree $n$ and let $\Gamma$ be the set of distinct Hecke characters of $\mathbb A_{K}^{\times}$ associated to this extension by class-field-theory. Now given a Hecke charactr $\chi$ of $\mathbb A_{K}^{\times}$, we define its base-change to $\mathbb A_{L}^{\times}$ by $\chi_L:=\chi \cdot \mathbb N_{L/K}$. Then one should check that: $$L(s,\chi_L)=\Pi _{\omega\in\Gamma }L(s,\chi\omega)$$

Conversely, suppose $L/K$ is cyclic with $Gal(L/K)=<\sigma >$. Let $\chi^\prime$ be a Hecke character of $\mathbb A_L^\times$ with $\chi^\prime=\chi^\prime\cdot\sigma$, then one might expect that $$\chi^\prime=\chi_L$$ for some Hecke character of $\mathbb A_K^{\times}$.

Automorphic-Induction

Let $L/K$ be an extension of number fields of degree $n$ and $\omega$ a Hecke character of $\mathbb A_L^\times$. Then there exists a partition $n=n_1+...+n_r$ and cuspidal automorphic representations $\pi_i$ of $GL_{n_i}$ such that $$L(s,\omega)=\Pi_{i=1}^rL(s,\pi_i)$$

My question is: How to prove these results ? Or tell me some reference. Please feel free to choose any one of questions to answer, not necessarily all.

Thank you very much.

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Where did you find the claim for the 2nd statement? Is this only a hope? The first statement is Artin reciprocity. Google will give you lots of references. – Marc Palm Aug 10 '12 at 11:24
Bump "Automorphic forms..." discusses some sort of base change in its first chapter, perhaps that helps. – Marc Palm Aug 10 '12 at 11:25
I think it is considered not appropriate to simultaneously post the same question here and on Math SE... – paul garrett Aug 10 '12 at 14:06