Compositio mathematica | |
Rank 2 local systems and abelian varieties II | |
article | |
Raju Krishnamoorthy1  Ambrus Pál2  | |
[1] Arbeitsgruppe Algebra und Zahlentheorie, Fakultät Mathematik und Naturwissenschaften, Bergische Universität Wuppertal;Department of Mathematics, Imperial College | |
关键词: F-isocrystals; local systems; abelian varieties; 11G10; 14D10; 14F30; 14G35; | |
DOI : 10.1112/S0010437X22007333 | |
学科分类:数学(综合) | |
来源: Cambridge University Press | |
【 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 $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$ .
【 授权许可】
CC BY
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