Working with Galois representations part1(Machine Learning 2024)
- Moduli stacks of Galois representations and the p-adic local Langlands correspondence for GL2(Qp)(arXiv)
Author : Christian Johansson, James Newton, Carl Wang-Erickson
Abstract : We give a categorical formulation of the p-adic local Langlands correspondence for GL2(Qp),as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of Gal(Q¯¯¯¯p/Qp). Moreover, we relate our version of the p-adic local Langlands correspondence for GL2(Qp) to the cohomology of modular curves through a local-global compatibility formula.
2. Constructing abelian varieties from rank 3 Galois representations with real trace field(arXiv)
Author : Raju Krishnamoorthy, Yeuk Hay Joshua Lam
Abstract : Let U/K be a smooth affine curve over a number field and let L be an irreducible rank 3 Q¯¯¯¯ℓ-local system on U with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that L extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking U, there exists an abelian scheme f:BU→U such that L is a summand of R2f∗Q¯¯¯¯ℓ(1). The key ingredients are: (1) the totally real assumption implies L admits a square root M; (2) the trace field of M is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over UK¯ geometrically realizing L; and (3) Deligne’s weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneratio
