Mod-p Galois representations not arising from abelian varieties

2020/10/31 by Shiva Chidambaram, Chidambaram, Shiva · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2011.00158

Abstract

It is known that any Galois representation ρ: G_ℚ → GL(2,\mathbbF_p) with determinant equal to the mod-p cyclotomic character, arises from the p-torsion of an elliptic curve over ℚ, if and only if p ≤ 5. In dimension g = 2, when p ≤ 3, it is again known that any Galois representation valued in GSp(4,\mathbbF_p) with cyclotomic similitude character arises from an abelian surface. In this paper, we study this question for all primes p and dimensions g ≥ 2. When g ≥ 2 and (g,p) ≠ (2,2), (2,3), (3,2), we prove the existence of a Galois representation over ℚ valued in GSp(2g,\mathbbF_p) with cyclotomic similitude character, that cannot arise as the p-torsion representation of any g-dimensional abelian variety over ℚ.

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