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In the theory of p-adic modular forms there is a certain construction called the Coleman-Mazur eigencurve. The spectral halo conjecture roughly states that if you remove a closed subdisc of the weight space, the eigencurve is an infinite disjoint union of finite flat covers of what remains of the weight space.

For a person who does not intrinsically care about the eigencurve, what interest does the halo conjecture pose? What corollaries does it have that do not directly involve the eigencurve?

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There was just a paper posted by Newton and Thorne (https://arxiv.org/abs/1912.11261) on automorphy of symmetric powers of eigenforms that might be something you care about. See also a recent blog post here: https://www.galoisrepresentations.com/2019/12/30/new-results-in-modularity-christmas-update-ii/.

Of course, you might also not care about that thing. Perhaps someone can answer you more usefully if you describe more carefully what you actually care about?

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  • $\begingroup$ how is the halo conjecture relevant to the arguments described in the paper? Would the theorem be easier (or harder) to prove if it were true? $\endgroup$
    – user145520
    Jan 10 '20 at 22:09
  • $\begingroup$ to be more precise: the halo conjecture is not directly mentioned in the paper. A description of the eigencurve in some specific case by Buzzard and Kilford is mentioned but that is too specific in my view. $\endgroup$
    – user145520
    Jan 11 '20 at 0:32
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    $\begingroup$ The theorem of Buzzard & Kilford is one of the few known cases of the halo conjecture. So while I agree that the halo conjecture is not directly mentioned, it needn't be because the authors only need one specific case of the conjecture, the one proven by Bzuzard & Kilford, to pull off their argument. (I'll add that there is a lot of other mathematics in this paper, all very interesting. The automorphy is reduced to special cases using the halo structure, but those special cases are themselves a big thing.) $\endgroup$
    – jfb
    Jan 11 '20 at 15:58

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