Work performed so far and main results achieved
Together with my collaborators, I have made significant progress towards the goals of the project.
1. In joint work with P. Scholze, I completed the proof of a vanishing theorem for the "generic part" of the cohomology of certain non-compact unitary Shimura varieties. This was mentioned in the proposal as work in progress and was completed during the first year of the project. This result makes unconditional my previous work on potential automorphy over CM fields, joint with P. Allen, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor and J. Thorne. This latter paper was revised for publication during the time I was supported by the ERC project.
2. In joint work with M. Emerton, T. Gee and D. Savitt, I constructed moduli stacks of two-dimensional mod p Galois representations. Recent conjectures of Zhu suggest a close connection between understanding the cohomology of Shimura varieties as in 1) and understanding coherent sheaves on moduli stacks of Galois representations.
3. In a series of two papers, that are joint work with D. Gulotta, C. Y. Hsu, C. Johansson, L. Mocz, E. Reinecke and S.C. Shih, and with D. Gulotta and C. Johansson, I proved a vanishing resuly for the cohomology with compact support of certain Shimura varieties with infinite level at p. The first paper aslo contains an application to the Galois representations associated to torsion in the cohomology of locally symmetric spaces for GL_n over CM fields.
4. In joint work with M. Tamiozzo, who was supported as a postdoctoral research associate by the ERC project, I proved a vanishing theorem for the cohomology of Hilbert modular varieties. This relies on studying the "generic part" of the cohomology but also obtains bounds beyond the generic case.
5. In joint work with J. Newton, I proved a local-global compatibility result for torsion in the cohomology of locally symmetric spaces in the crystalline case. This was generalised to the potentially semi-stable case by my PhD student, B. Hevesi, who was supported as a research assistant by the ERC project.
6. Newton and I used our local-global compatibility result in the crystalline case and an intermediate result from point 2 above to prove that 100% of elliptic curves over a fixed imaginary quadratic field are modular. In particular, we were able to show that all elliptic curves defined over the Gaussian numbers Q(i) are modular.
