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.
2 thoughts on “Elliptic curves over Q(i) are potentially automorphic”
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?
The “modularity of a class of ellitpic curves” you are referring to, I assume is this famous business with Fermat’s Last Theorem and so on. This later picture is an example of the Langlands Correspondence. In particular, it relates modular forms on the upper half plane modulo some arithmetic subgroup to Galois representations associated to Ellitpic curves defined over Q. The general Langlands conjectures involves (of course) more general objects such as ellitpic curves defined over an arbitrary number field and more strange analytic spaces than the upper half plane. In particular, the automorphy of an ellitpic curve over Q(i) means that you have shown that Galois representations associated to an ellitpic curve over Q(i) comes from an automorphic forms, (Generalizations of modular forms to these more general analytic spaces.) Thus, in the framework of Langalnds, it is a straightforward generalization. The importance of this result is that it is the first instance of proving a statement like this in which the associated analytic spaces do not carry an algebraic structure. (The upper half plane modulo an arithemetic group will be defined as an algebraic variety over some curve, whose complex points are the plane you are familiar with.) This makes them not at all amenable to many of the methods that one would use in proving a statement such as the modularity of elliptic curves.
Hope this answers yours question.