What is the best known result concerning the existence of companion forms for classical modular forms? Gross' tameness criterion paper is always mentioned with a "unchecked compatibility" caveat? Are the results in this paper bona-fide theorems? The Coleman-Voloch paper on the topic (which does not rely on these unchecked compatibilities) seems not to allow k=2 (but Gross' paper is fine with k=2). Does Khare-Winterberger's proof of Serre's conjecture trump all of this? If so, is Serre's conjecture known unconditionally now or are there still unresolved cases?
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Companion forms
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