Rank 2 local systems and abelian varieties II

[摘要] 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 $U\subset X$ 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 $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$ .

[学科分类] 数学（综合）

[关键词] F-isocrystals;local systems;abelian varieties;11G10;14D10;14F30;14G35