论文标题
等级2本地系统和Abelian品种II
Rank 2 local systems and abelian varieties II
论文作者
论文摘要
令$ x/\ mathbb {f} _ {q} $成为平滑，几何连接的，准注射量的多样性。令$ \ Mathcal {e} $为$ x $上的半完整$ f $ -isocrystal。假设$ \ mathcal e $的不可约合$ \ mathcal {e} _i $具有等级2，确定性$ \ bar {\ mathbb {q}} _ p（-1）$，而无限单型$。假设进一步，对于$ x $的每个关闭点$ x $，$ \ mathcal {e} $ at $ x $的特征多项式位于$ \ m athbb {q} [q} [t] \ subset \ subset \ mathbb q_p [t] $中。然后存在一个非平凡的开放集$ u \子集X $，因此$ \ Mathcal {e} | _U $来自$ u $的Abelian品种。 作为一个应用程序，让$ l_1 $为不可约的Lisse $ \ bar {\ Mathbb {q}} _ l $ sheaf，$ x $上的$ x $，具有排名2的$ \ bar {\ mathbb {q}} _ l（q}} _ l（-1）$，infinite monodromy in $ \ iffty $。然后，所有的结晶伴侣都存在$ l_1 $的所有（如Deligne的Crystalline Companions的预测）时，并且仅在存在非平凡的开放式套装$ u \ u \ subset x $和Abelian方案$π_U\ COLON A_U \ COLON A_U \ cOLON A_U \ rightarrow U $中，因此$ r^1（π_U）_*\ bar {\ mathbb {q}}} _ l $。
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 $π_U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^1(π_U)_*\bar{\mathbb{Q}}_l$.
