Restriction of Hecke Characters

Asked Feb 05 '21 at 17:57

Active Feb 05 '21 at 17:57

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Let $L$ be a number field and let $\Psi$ be a finite order Hecke character on $L$ with (finite part) of conductor as $\mathfrak{f}$. Suppose we define $$\psi(m) :=\Psi(m\mathcal{O}_L)$$ for all integers $m\in\mathbb{Z}$. Does $\psi$ define a Dirichlet character? If so what is the modulus?

I kind of feel that this should be true and the modulus should be $\mathfrak{f}\cap\mathbb{Z}$ but I don't know how to go about it.

Thank you in advance.

asked Feb 05 '21 at 17:57

Krishnarjun

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It might be simpler to think about idele-class-group characters restricted to a diagonally imbedded idele-class-group of subfield. – paul garrett Feb 05 '21 at 18:09
Is this the embedding you are talking about? $\mathbb{Q}^\times\hookrightarrow L^\times \hookrightarrow \mathbb{A}_L^\times$ ? – Krishnarjun Feb 05 '21 at 18:11
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Yes. To my taste, it gives a cleaner picture of the local info. – paul garrett Feb 05 '21 at 18:13
I shall try to write down the details, thank you for the suggestion... :) – Krishnarjun Feb 05 '21 at 18:14

Restriction of Hecke Characters

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