This spectacular theorem was announced by Richard Taylor on Thursday, in a lecture at the joint meetings. Taylor credited this result and others to Allen-Calegari-Caraiani-Gee-Helm-Le Hung-Newton-Scholze-Taylor-Thorne (!), as an outcome of the (not so) secret mini-conference which took place at the IAS this fall. The key new input here is work in progress of Caraiani-Scholze on the cohomology of non-compact unitary Shimura varieties, which can be leveraged to check (at least in some cases) the most difficult hypothesis in the Calegari-Geraghty method: local-global compatibility at l=p for torsion classes.

The slides from my talk can be found here. Naturally I managed to say “diamond” a bunch of times.

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As a mathematician who is not an algebraic geometer or number theorist, but who has an amateur interest in such things, how does this relate to modularity of a class of elliptic curves?

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