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Constructing abelian varieties from rank 2 Galois representations
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 4 / April 2024
- Published online by Cambridge University Press:
- 07 March 2024, pp. 709-731
- Print publication:
- April 2024
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- Article
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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 lisse 
$\overline {{\mathbb {Q}}}_\ell$-sheaf on 
$U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field 
$E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some closed point 
$x$ of 
$X\setminus U$. 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 and Tsimerman, who show that when 
$E=\mathbb {Q}$, then 
${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve 
$E_U\rightarrow U$.
