Tales from the dropbox

As today’s service to the number theory entertainment complex, I could write an ill-advised rant about the abc conjecture situation. Instead, here’s another dropbox link. This time it’s some notes I wrote while trying to understand Kohlhaase’s paper Smooth duality in natural characteristicWarning: These are only about 40% done. The only original content (so far) is Proposition 1.6 and Theorem 1.12.

Elliptic curves over Q(i) are potentially automorphic

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.