Lifting Subgroups of Symplectic Groups over ℤ / ℓ ℤ

2016/07/16 by Aaron Landesman, Ashvin Swaminathan, Landesman, Aaron +5
Mathematics · #11G10 #20G25 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1607.04698

openalex publication_date 2016/07/16 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28

Abstract

For a positive integer g, let Sp_2g(R) denote the group of 2g × 2g symplectic matrices over a ring R. Assume g ≥ 2. For a prime number ℓ, we give a self-contained proof that any closed subgroup of Sp_2g(ℤ_ℓ) which surjects onto Sp_2g(ℤ/ℓℤ) must in fact equal all of Sp_2g(ℤ_ℓ). The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.

Citations

• An explicit Jacobian of dimension 3 with maximal Galois action

Related

• Some remarks on the symplectic group Sp(2g, ℤ)
• Lifting images of standard representations of symmetric groups
• Siegel Modular Varieties and the Eisenstein Cohomology of \PGL_2g+1
• Unisingular subgroups of symplectic group Sp_2n(2) for 2n<250
• Symplectic representations of inertia groups