4
$\begingroup$

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.

$\endgroup$
3
  • 1
    $\begingroup$ 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. $\endgroup$
    – Marc Palm
    Aug 10, 2012 at 11:24
  • $\begingroup$ Bump "Automorphic forms..." discusses some sort of base change in its first chapter, perhaps that helps. $\endgroup$
    – Marc Palm
    Aug 10, 2012 at 11:25
  • 3
    $\begingroup$ I think it is considered not appropriate to simultaneously post the same question here and on Math SE... $\endgroup$ Aug 10, 2012 at 14:06

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.