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What is known about modularity for $\mathrm{GL}_2/\mathbb{Q}(\sqrt[3]{2})$?

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    $\begingroup$ Welcome to MO! Some background might be useful here. It never hurts to include more details or at least some keywords, especially when there is nothing easy to look up (I have no idea how to find out what your question means even with Google at my disposal). For example, is $\operatorname{GL}_2$ here referring to $\operatorname{GL}_2(\mathbb{Q})$? Or $\operatorname{GL}_2(\mathbb{Z})$? Etc. What does the notation $\operatorname{GL}_2 / \mathbb{Q}(\sqrt[3]{2})$ mean? Is it an elliptic curve? Etc. Once the objects are defined, you don't have to define e.g. modularity (this can then be looked up). $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Sep 12, 2021 at 12:17
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    $\begingroup$ @Carl-FredrikNybergBrodda I'm pretty sure this refers to modular forms on the algrebraic group $GL_{2, \mathbb Q(\sqrt[3]{2})}$, i.e functions on $GL_2 ( \mathbb A_{\mathbb Q( \sqrt[3]{2} )} )/ GL_2 ( \mathbb Q(\sqrt[3]{2}))$. $\endgroup$
    – Will Sawin
    Commented Sep 12, 2021 at 13:08
  • $\begingroup$ @WillSawin OK, thanks! $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Sep 12, 2021 at 13:15
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    $\begingroup$ As for the question, I think that because this field is neither totally real nor CM, almost nothing is known, but I"m not really an expert... $\endgroup$
    – Will Sawin
    Commented Sep 12, 2021 at 13:40
  • $\begingroup$ Will Sawin is right: essentially nothing is known. $\endgroup$
    – David Hansen
    Commented Sep 14, 2021 at 14:54

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