Applications to modularity of recent progress on the cohomology of Shimura varieties

Organizers: Ana Caraiani and Richard Taylor

Participants: Patrick Allen, Frank Calegari, Ana Caraiani, Matt Emerton, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack Thorne

This Working Group was designed to take advantage of work in progress of Ana Caraiani and Peter Scholze on the vanishing below the middle degree of (much of the) the cohomology of a Shimura variety to tackle questions on modularity, such as the (potential) modularity of elliptic curves over imaginary quadratic fields. 11 mathematicians were invited to participate. The first two days were dedicated to fairly traditional lectures on recent developments which we thought might be relevant. These lectures were prepared ahead of time. Some of the lectures were polished presentations, while others were more speculative of the sort one might not like to present to a large audience. The third day was devoted to a discussion in the whole group with a series of participants going to the blackboard for varying periods of time. This seems to have led to key progress. On the fourth day the working group split into 3 subgroups each of which worked on sub-problems which had been identified. Twice during the day the whole working group reconvened to share progress and align goals. The fifth day was largely devoted to taking stock of what had been achieved and to planning the writing of a joint paper. In order to encourage the sharing of more speculative ideas, outside participation in the lectures was not encouraged.

__Outcomes__

A Caraiani Lecture Reciprocity laws for torsion classes

F Calegari Lecture Counting Galois representations