arXiv 1012.32232010-12-15math.NT

函数域的环面自守形式

Toroidal automorphic forms for function fields

Oliver Lorscheid

备注：: 32 pages

摘要：

空间的环形自守形式介绍了Zagier在1979年。设$F$为全局字段。$\GL(2)$上的自守形式是环面的，如果它沿所有嵌入的非分裂环面具有消失的常傅立叶系数。在这个空间的兴趣源于这样一个事实（除其他外），即重量$s$的爱森斯坦级数是环形的，如果$s$是zeta函数的非平凡零点，因此与黎曼假设建立了联系。在本文中，我们集中在函数域的情况下。我们显示以下结果。一个非平凡的重量$s$和Hecke字符$χ$的爱森斯坦级数的$(n-1)$阶导数是环形的，当且仅当$L(χ,s+1/2)$在$s$中为零，以至少订购$n$（对于“仅当”-部分，我们假设$F$的特征是奇数）。Eisenstein级数不存在非平凡的环面剩余。如果特征不为2，则非分歧艾森斯坦级数的导数空间的维数等于$h(g-1)+1$;在特征2中，维数由这个数从下到上有界。这里$g$是属，$h$是$F$的类号。环面自守形式空间是一个容许表示，且每个不可约子商都是调和的。

英文摘要：

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let $F$ be a global field. An automorphic form on $\GL(2)$ is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight $s$ is toroidal if $s$ is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established. In this paper, we concentrate on the function field case. We show the following results. The $(n-1)$-th derivative of a non-trivial Eisenstein series of weight $s$ and Hecke character $χ$ is toroidal if and only if $L(χ,s+1/2)$ vanishes in $s$ to order at least $n$ (for the "only if"-part we assume that the characteristic of $F$ is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals $h(g-1)+1$ if the characterisitc is not 2; in characteristic 2, the dimension is bounded from below by this number. Here $g$ is the genus and $h$ is the class number of $F$. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.

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