Strongly compatible systems associated to semistable abelian varieties

2025/05/04 by Mark Kisin, Kisin, Mark, Zhou Rong +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #advanced mathematical theories #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2505.02165

Abstract

We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given an abelian variety A over a number field E⊂ \mathbb C, we prove that after replacing \mathbb E by a finite extension, the action of Gal(\mathrm E/\mathrm E) on the ℓ-adic cohomology \mathrm H¹_\mathrm\acuteet(A_\mathrm E,\mathbb Q_ℓ) gives rise to a strongly compatible system of ℓ-adic representations valued in the Mumford--Tate group \mathbf G of A. This involves an independence of ℓ-statement for the Weil--Deligne representation associated to A at places of semistable reduction, extending previous work of ours at places of good reduction.

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