Rank 2 local systems and abelian varieties II

Abstract

Let $X/{\mathbb{F}}_q$ be a smooth, geometrically connected, quasiprojective variety. Let $\mathcal{E}$ be a semisimple 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 non-trivial open set $U\subset X$ 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 exists a non-trivial open set $U\subset X$ and an abelian scheme ${\pi }_U:A_U\to U$ such that $L_1{|}_U$ is a summand of $R^1({\pi }_U{)}_{\ast }{\bar{\mathbb{Q}}}_l$.

21 pages, comments very welcome! V2: slightly reorganized, typos fixed. v3: substantial revision, in response to referee comments

References in corpus (1)

• Rank 2 local systems and abelian varieties