Lubin-Tate and multivariable (φ,O_K^×)-modules in dimension 2

2024/03/30 by Yitong Wang, Wang, Yitong
Mathematics · #Algebraic structures and combinatorial models #Holomorphic and Operator Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2404.00396

Abstract

Let p be a prime number, K a finite unramified extension of ℚ_p and \mathbbF a finite extension of \mathbbF_p. For ρ any reducible two-dimensional representation of Gal(K/K) over \mathbbF, we compute explicitly the associated étale (φ,O_K^×)-module D_A^⊗(ρ) defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let π be an admissible smooth representation of GL₂(K) over \mathbbF occurring in some Hecke eigenspaces of the mod p cohomology and ρ be its underlying two-dimensional representation of Gal(K/K) over \mathbbF. Assuming that ρ is maximally non-split, we prove under some genericity assumption that the associated étale (φ,O_K^×)-module D_A(π) defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to D_A^⊗(ρ). This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where ρ was assumed to be semisimple.

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