arXiv 1012.35132010-12-16math.NTmath.AG

Hecke算子图

Graphs of Hecke operators

Oliver Lorscheid

备注：: 36 pages

摘要：

设$X$是$\F_q$上的曲线，函数域为$F$。在本文中，我们定义了一个图的每一个Hecke算子与固定分歧。先验地，这些图可以被看作是一种方便的语言来组织Hecke算子对自守形式的作用公式。然而，他们将被证明是一个强大的工具，显式计算和证明有限维的结果。我们发展了一个结构理论的某些图$G_x$的unramified Hecke运营商，这是一个类似的静脉塞尔的理论的分支的Bruhat山雀树。准确地说，$G_x$是局部的一个商的Bruhat山雀树，并有许多组件。通过对$G_x$在$X$上秩为2的丛的解释和约化理论的方法，证明了$G_x$是许多尖点和核的结合，尖点是简单性质的无限子图，核是有限子图，它严重依赖于$F$的算术.我们描述了如何恢复unramified自守形式的功能图$G_x$。在尖点和环形条件的示例性情况下，我们展示了如何在$G_x$上的函数的线性条件导致有限维的结果。特别地，我们重新获得了非分歧尖点形式空间和非分歧环面自守形式空间的有限维性。在附录中，我们计算了各种各样的例子，图在有理函数域。

英文摘要：

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.

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