Modularity lifting beyond the Taylor–Wiles method
By Frank Calegari, David Geraghty. 2012
109 citations recorded by the source corpus.
We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions—one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side, applies to automorphic forms on the group $$\mathrm {GL}(n)$$GL(n) over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato–Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of $$\mathrm {GL}(2)$$GL(2) over $$\mathbf {Q}$$Q, we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight 1. Such algebras may well contain p-torsion. Moreover, we also completely solve the problem (for p odd) of determining the multiplicity of an irreducible modular representation $$\overline{\rho }$$ρ¯ in the Jacobian $$J_1(N)$$J1(N), where N is the minimal level such that $$\overline{\rho }$$ρ¯ arises in weight two.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0b6cd0ccf59f923ad04bc62375c76e3ace79ab3b
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Potential automorphy over CM fields (2018, methodology): “out a version of the ‘Ihara avoidance’ trick under somewhat weaker assumptions than those used in [CG12].”
- Potential Automorphy and the Leopoldt conjecture (2014, methodology): “e expected dimension of these deformation rings, which in turn implies the formula (1.1) for the codimension of the cohomology as Λ-module. We take as our starting point the work of Calegari–Geraghty [CG] showing how to make the Taylor– Wiles method work in the ‘positive defect’ context, assuming th”
- A torsion Jacquet--Langlands correspondence (2012, background): “A recent approach to circumventing these obstructions can be found in the work of the first author and David Geraghty [21].”
- Derived Galois deformation rings (2016, methodology): “sed to upgrade this action to a graded action of π˚Rcrys. To carry out this upgrading, we establish an isomorphism between π˚Rcrys and a limit ring that arises in the Taylor-Wiles method (in the form [6] of Calegari-Geraghty). Ourresultsrequire someassumptions (Conjecture 6.1,similartotheconjectures”
- On locally analytic vectors of the completed cohomology of modular curves (2020, result): “. Remark 1.0.6. This result was already known by the work of Pilloni-Stroh [PS16]. Before their work, much work on this problem was done in the ordinary case: Buzzard-Taylor [BT99], Calegari-Geraghty [CG18]... and has been generalized to the case of Hilbert modular forms. I hope this paper gives a n”
- Strata Hasse invariants, Hecke algebras and Galois representations (2015, methodology): “st in such “torsion Langlands correspondences” has grown considerably due to the pivotal role that they play in the CalegariGeraghty program of pushing the Taylor-Wiles method beyond the regular case [15]. Scholze achieved a breakthrough in the Betti case, by associating pseudo-representations to th”