I just came across this paper which, judging by what I understood, establishes the Langlands reciprocity conjecture for a certain Shimura variety. My question, regardless of the validity of the proof, is: would such a result have far-reaching consequences in number theory? If so, what would they be?
Thanks in advance.
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asked May 13, 2015 at 23:09
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5$\begingroup$ From reding the first page of the paper I can say that it is rubbish. The author speaks of Shimura varieties attached to reductive groups, but not all groups have such, he introduces convolution on $L^2(\Gamma\backslash G)$ which is not well-defined and so on. $\endgroup$– user1688Commented May 14, 2015 at 7:22
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4$\begingroup$ @Corbennick The author has a record of taking genuine stuff from Cstar algebras (irrational tori, etc) and then coming up with grander and grander window-dressing. (I'm not qualified to judge the correctness or otherwise of said window-dressing.) $\endgroup$– Yemon ChoiCommented May 14, 2015 at 10:47
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2$\begingroup$ It seems to me that the indicated preprint gets so far off the rails so quickly that it's hard to visualize specific details of a plausible, corrected version. E.g., the "convolution" product at the bottom of the first page certainly does not converge, ... but there are many related integrals which do, but, ... And, yes, as @Corbennick comments, not all reductive groups have arithmetic quotients which are algebraic varieties... So the notion of a "corrected" version is too ambiguous to discuss, I fear. $\endgroup$– paul garrettCommented May 14, 2015 at 16:46
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