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- W4282959650 abstract "Let $X/mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $mathcal {E}$ be a semi-simple overconvergent $F$ -isocrystal on $X$ . Suppose that irreducible summands $mathcal {E}_i$ of $mathcal {E}$ have rank 2, determinant $bar {mathbb {Q}}_p(-1)$ , and infinite monodromy at $infty$ . Suppose further that for each closed point $x$ of $X$ , the characteristic polynomial of $mathcal {E}$ at $x$ is in $mathbb {Q}[t]subset mathbb {Q}_p[t]$ . Then there exists a dense open subset $Usubset X$ such that $mathcal {E}|_U$ comes from a family of abelian varieties on $U$ . As an application, let $L_1$ be an irreducible lisse $bar {mathbb {Q}}_l$ sheaf on $X$ that has rank 2, determinant $bar {mathbb {Q}}_l(-1)$ , and infinite monodromy at $infty$ . Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exist a dense open subset $Usubset X$ and an abelian scheme $pi _Ucolon A_Urightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(pi _U)_*bar {mathbb {Q}}_l$ ." @default.
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- W4282959650 date "2022-04-01" @default.
- W4282959650 modified "2023-09-30" @default.
- W4282959650 title "Rank 2 local systems and abelian varieties II" @default.
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- W4282959650 doi "https://doi.org/10.1112/s0010437x22007333" @default.
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