Constructing abelian varieties from rank 2 Galois representations

2022/08/03 by Krishnamoorthy, Raju, Yang, Jinbang, Zuo, Kang
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2208.01999

Abstract

Let U be a smooth affine curve over a number field K with a compactification X and let \mathbb L be a rank 2, geometrically irreducible \mathbb Q_ℓ-local system on U with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field E⊂ \mathbb Q_ℓ, and has bad, infinite reduction at some closed point x of X∖ U. We show that \mathbb L occurs as a summand of the cohomology of a family of abelian varieties over U. The argument follows the structure of the proof of a recent theorem of Snowden-Tsimerman, who show that when E=\mathbb Q, then \mathbb L is isomorphic to the cohomology of an elliptic curve E_U→ U.

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