Periodic Reporting for period 4 - PariTorMod (P-adic Arithmetic Geometry, Torsion Classes, and Modularity)
Período documentado: 2023-07-01 hasta 2024-10-31
Summary of the context and overall objectives of the project
The overall theme of this project is the interplay between p-adic arithmetic geometry and the Langlands correspondence over number fields. The Langlands program is a "grand unified theory of mathematics", an intricate network of conjectures whose goal is to connect number theory to other areas of mathematics, such as representation theory and harmonic analysis. At the heart of the Langlands program lies reciprocity, the conjectural correspondence between Galois representations and automorphic forms. The Galois to automorphic direction of this correspondence was essentially wide open until a quarter century ago, when Wiles's proof of Fermat's last theorem led to a blossoming of the area and to the construction of many new reciprocity laws.
The first wave of results originated in the Taylor-Wiles method for proving modularity lifting theorems. We are now experiencing a second wave of reciprocity laws, initiated by Calegari and Geraghty, who proposed a decade ago a method for proving modularity beyond the Taylor-Wiles setting. Their method relies on deep conjectures about torsion in the cohomology of locally symmetric spaces. The overall goal of this project is to capitalise on recent progress in p-adic arithmetic geometry to estalish new instances of the Langlands correspondence in the setting of locally symmetric spaces for GL_n over CM fields, via the Calegari-Geraghty method.
This project has three interrelated strands: the first concerns the geometry and cohomology of Shimura varieties, the second concerns local-global compatibility for the Galois representations associated to torsion in the cohomology of locally symmetric spaces, and the third concerns modularity lifting theorems and applications.
The overall theme of this project is the interplay between p-adic arithmetic geometry and the Langlands correspondence over number fields. The Langlands program is a "grand unified theory of mathematics", an intricate network of conjectures whose goal is to connect number theory to other areas of mathematics, such as representation theory and harmonic analysis. At the heart of the Langlands program lies reciprocity, the conjectural correspondence between Galois representations and automorphic forms. The Galois to automorphic direction of this correspondence was essentially wide open until a quarter century ago, when Wiles's proof of Fermat's last theorem led to a blossoming of the area and to the construction of many new reciprocity laws.
The first wave of results originated in the Taylor-Wiles method for proving modularity lifting theorems. We are now experiencing a second wave of reciprocity laws, initiated by Calegari and Geraghty, who proposed a decade ago a method for proving modularity beyond the Taylor-Wiles setting. Their method relies on deep conjectures about torsion in the cohomology of locally symmetric spaces. The overall goal of this project is to capitalise on recent progress in p-adic arithmetic geometry to estalish new instances of the Langlands correspondence in the setting of locally symmetric spaces for GL_n over CM fields, via the Calegari-Geraghty method.
This project has three interrelated strands: the first concerns the geometry and cohomology of Shimura varieties, the second concerns local-global compatibility for the Galois representations associated to torsion in the cohomology of locally symmetric spaces, and the third concerns modularity lifting theorems and applications.
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.
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.
Progress beyond the state of the art and expected results until the end of the project
The results on non-compact unitary Shimura varieties, on local-global compatibility, and on the modularity of elliptic curves over imaginary quadratic fields went significantly beyond the state of the art in the field. These represent progress along all three main strands of the project.
The results for Shimura varieties with infinite level at p represent an unexpected development; they point to a parallel development to the one in project 1 with "generic" at a prime l different from p replaced with "ordinary" at p. There is also an unexpected connection with the higher Hida and Coleman theories being developed by Boxer and Pilloni. The results on Hilbert modular varieties also represent an unexpected development, because we employ a new technique, based on geometric Jacquet-Langlands functoriality for Igusa varieties.
The results on non-compact unitary Shimura varieties, on local-global compatibility, and on the modularity of elliptic curves over imaginary quadratic fields went significantly beyond the state of the art in the field. These represent progress along all three main strands of the project.
The results for Shimura varieties with infinite level at p represent an unexpected development; they point to a parallel development to the one in project 1 with "generic" at a prime l different from p replaced with "ordinary" at p. There is also an unexpected connection with the higher Hida and Coleman theories being developed by Boxer and Pilloni. The results on Hilbert modular varieties also represent an unexpected development, because we employ a new technique, based on geometric Jacquet-Langlands functoriality for Igusa varieties.