论文标题
等级两个本地系统的痕迹场的界限
Boundedness of trace fields of rank two local systems
论文作者
论文摘要
让$ p $是固定的质子数，而$ q $ a Power a $ p $。对于$ \ mathbb {f} _q $及其上的任何本地系统的任何曲线，我们在封闭点（称为跟踪字段）的Frobenii痕迹生成了一个数字字段。我们表明，随着我们在特征$ p $中的所有尖曲线（g，n）$ p $ and Angrove范围内，并对满足无限条件的两个本地系统进行排名，因此在$ p $且有限程度的痕迹范围内均未受到影响。这证明了通过数值计算获得的Kontsevich的观察结果，这反过来又与Maeda对功能字段的猜想的类似物密切相关。证据的关键要素是Chin的单元组$ \ ell $独立性的定理，以及$ \ mathrm {gl} _2 $ type的亚伯语方案的界限，使用部分hasse novariants获得了积极特征的曲线；后者是Faltings的Arakelov定理的Abelian品种的类似物。
Let $p$ be a fixed prime number, and $q$ a power of $p$. For any curve over $\mathbb{F}_q$ and any local system on it, we have a number field generated by the traces of Frobenii at closed points, known as the trace field. We show that as we range over all pointed curves of type $(g,n)$ in characteristic $p$ and rank two local systems satisfying a condition at infinity, the set of trace fields which are unramified at $p$ and of bounded degree is finite. This proves observations of Kontsevich obtained via numerical computations, which are in turn closely related to the analogue of Maeda's conjecture over function fields. The key ingredients of the proofs are Chin's theorem on independence of $\ell$ of monodromy groups, and the boundedness of abelian schemes of $\mathrm{GL}_2$-type over curves in positive characteristics, obtained using partial Hasse invariants; the latter is an analogue of Faltings' Arakelov theorem for abelian varieties in our setting.
