### Galois Representations

During both terms of the 2010-11 year, the School of Mathematics held a special program on Galois Representations and Automorphic Forms. The program was organized by the School’s Distinguished Visiting Professor, Richard Taylor of Harvard University.

The program had about 33 long-term participants and many other people visited for shorter periods. Six mini-courses were organized (41 hours in all), two weekly seminars (34 hours in total) and a one-week workshop (18 one-hour talks). In addition, participants in the program took advantage of the joint IAS/PU number theory seminar and the IAS members’ seminar. All these activities (except the workshop and members’ seminar) took place on Wednesday afternoons and Thursdays. This allowed the participants to concentrate on their own work for the rest of the week and also made it easier for participants from neighboring universities to attend our events.

The long-term participants in the program were: F. Calegari, L. Clozel, P. Colmez, J.-F. Dat, J.-M. Fontaine, A. Ganguli, D. Geraghty, T. Haines, M. Harris, F. Herzig, A. Jorza, T. Kaletha, C. Khare, K.-W. Lan, R. Liu, T. Liu, E. Mantovan, S. Morel, J. Newton, W. Niziol, Y. Sakellaridis, S.-W. Shin, C. Skinner, R. Taylor, Y. Tian, E. Urban, J. Weinstein and A. Wiles. In addition, M. Emerton and P. Scholze both made more than one visit, and there were several students from Princeton University who attended regularly. Students involved with program activities from outside the Princeton area were L. Chung, A. Caraiani, W. Goldring, B. Le Hung and Jack Thorne.

In the fall P. Colmez gave a 14-hour mini-course on his work on the p-adic local Langlands correspondence, F. Calegari gave a 3-hour mini-course on completed cohomology, and M. Emerton gave a 3-hour mini-course on local-global compatibility in the p-adic Langlands program. In the spring C. Skinner gave an 8-hour mini-course on his work with Urban on the Iwasawa main conjecture for GL(2), Philip Griffiths gave a 5-hour mini-course on automorphic cohomology, and D. Geraghty and R. Taylor gave an 8-hour mini-course on recent advances in automorphy/potential automorphy of Galois representations.

Of the 34 seminars given during the year, particular highlights of the seminar program were J. Weinstein’s beautiful breakthrough on semi-stable models for modular curves of all levels and P. Scholze’s exciting new take on the local Langlands conjecture for GL(n).

The talks in the workshop were uniformly of an extremely high standard. Speakers were selected to ensure talks about what was thought to be the most interesting recent developments. Despite the fact that the School made no special effort to invite young people to speak, it was very gratifying that of the 18 speakers 3 were graduate students, 5 were post-docs, while only 8 had tenure. The workshop was also very well attended with well over 100 participants. Some of the most exciting developments reported on at the workshop was the work of I. Pilloni, Stevens, Stroh et al on eigenvarieties for higher dimensional Shimura varieties, P. Scholze’s introduction of perfectoid spaces and his use of them to both re-prove p-adic comparison theorems and to attack the weight-monodromy conjecture, the exciting ideas of Calegari and Geraghty on proving modularity lifting theorems over imaginary quadratic fields (and probably other settings, where the original approach seemed to break down), and Kakde’s proof of the non-abelian main conjecture for characters over totally real fields.

Some breakthroughs that were announced and discussed were J. Weinstein’s initial breakthrough on the old (about 40 years) problem of finding semi-stable models for all modular curves took place before the start of the special year. However Weinstein had extensive discussions with other members of the program, and his results have taken on a more natural form (for example, more systematically passing to a limit over all level structures), and he has begun to generalize them beyond GL(2).

F. Calegari and D. Geraghty, who were both here for the whole year, have announced a breakthrough on modularity lifting theorems. All modularity lifting theorems to date only work in the `regular, odd, conjugate self-dual’ case, where the universal deformation ring and the Hecke algebras are as large as possible. This seemed to be a major restriction on the Taylor-Wiles method. (It is somehow easier to prove two things are equal when they are both as big as they can possibly be.) However, Calegari and Geraghty, working together during the special year) appear to have found a way to treat other cases. At the moment their results are restricted to some (important) special cases, but it seems plausible that this is just the start.

WORKSHOP TALKS:

L. Clozel “The Proof of the Burger-Sarnak Conjecture”

T. Kaletha “Supercuspidal L-packets”

V. Pilloni “Construction of Eigenvarieties and Coherent Cohomology”

A. Iovita “An overconvergent Eichler-Shimura map”

A. Caraiani “Local-global compatibility and monodromy”

S. Morel “Intersection cohomology is useless”

P. Scholze “Perfectoid spac”

J. Weinstein “A Semistable Model for the Tower of Modular Curves”

F. Calegari “Minimal Modularity Lifting Theorems for Imaginary Quadratic Fields”

M. Emerton “p-Adic Hodge-Theoretic Properties of Etale Cohomology with mod p Coefficients, and the Cohomology of Shimura Varieties – 1”

T. Gee “p-Adic Hodge-Theoretic Properties of Etale Cohomology with mod p Coefficients, and the Cohomology of Shimura Varieties – 2”

D. Geraghty “Local-Global Compatibility at Primes Dividing l”

M.-F. Vigneras “From General Etale (φ,Γ)-Modules to Representations of G(Qp)”

V. Paskunas “The Image of Colmez’s Montreal Functor”

G. Dospinescu “Locally Algebraic Vectors in the p-Adic Langlands Correspondence”

H. Hida “Vanishing of the μ-Invariant of p-Adic Hecke L Functions”

D. Burns “Recent Developments in Non-Commutative Iwasawa Theory – 1”

M. Kakde “Recent Developments in Non-Commutative Iwasawa Theory – 2”