Companion forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:48:47Z http://mathoverflow.net/feeds/question/65583 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65583/companion-forms Companion forms akula 2011-05-20T19:41:16Z 2011-05-20T23:57:26Z <p>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?</p> http://mathoverflow.net/questions/65583/companion-forms/65588#65588 Answer by Kevin Buzzard for Companion forms Kevin Buzzard 2011-05-20T20:53:04Z 2011-05-20T20:53:04Z <p>It might all depend on precisely what you mean by Serre's conjecture. Various versions are in print. Serre's original conjecture stayed away from \$k=1\$ and K-W resolved this version of the conjecture completely I believe, including all companion forms issues. Did you take a look at the K-W papers? They surely give a precise statement of what they prove. Edixhoven made the most optimistic conjecture, allowing weight 1, and I might be wrong but at the back of my mind I suspect that in some cases where the mod \$p\$ representation is trivial on a decomposition group at \$p\$, K-W only produce a weight \$p\$ form whereas what is conjectured is that there's a mod \$p\$ weight 1 form (in the sense of Katz).</p> http://mathoverflow.net/questions/65583/companion-forms/65596#65596 Answer by Felipe Voloch for Companion forms Felipe Voloch 2011-05-20T21:23:59Z 2011-05-20T21:30:52Z <p>In addition to what Kevin already said, there is some work of Bryden Cais and of Gabor Wiese that have removed some and perhaps all of the additional hypotheses. I can't guarantee for sure that it's all done, you should ask them. See <a href="http://arxiv.org/abs/1102.2302" rel="nofollow">http://arxiv.org/abs/1102.2302</a> and <a href="http://www.math.wisc.edu/~cais/Papers/PhDThesis/PhD.html" rel="nofollow">http://www.math.wisc.edu/~cais/Papers/PhDThesis/PhD.html</a></p> http://mathoverflow.net/questions/65583/companion-forms/65607#65607 Answer by A. Pacetti for Companion forms A. Pacetti 2011-05-20T23:57:26Z 2011-05-20T23:57:26Z <p>There is also a paper of Toby Gee which, as in K-W, uses tools of modularity lifting theorems. He proves the existence of a weight p form as mentioned by Kevin which works as well for Hilbert modular forms. You can check the arxiv paper <a href="http://arxiv.org/abs/math/0507507" rel="nofollow">http://arxiv.org/abs/math/0507507</a></p>