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What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
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Does the image of an p-adic Galois representation always lie in a finite extension?
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Does Ribet’s level lowering theorem hold for prime powers?
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What is the nature of the locus in the eigencurve associated to some conditions on the associated automorphic representation (at $p$)?