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Dec 2, 2019 at 19:51 comment added Joe Silverman @user125639 Sorry, I didn't mean this as an answer to the question in your last paragraph. I have no idea if or how this approach can be applied to the GL$_n$ case for $n\ge2$. I was just answering your initial request for a reference where the GL$_1$ argument appears.
Dec 2, 2019 at 19:11 comment added user125639 Thanks! This certainly answers the question, though the proof is more "direct" than what I had in mind in my last paragraph.
Dec 2, 2019 at 19:09 vote accept user125639
Dec 1, 2019 at 23:23 history answered Joe Silverman CC BY-SA 4.0