Shimura varieties for unitary groups and the doubling method
Shimura varieties for unitary groups and the doubling method
复制标题
酉群的志村品种和倍增法
DOI：
10.1007/978-3-030-68506-5_6
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Harris, Michael
中科院分区：
文献类型：
--
作者：
Harris, Michael
gl (n)的自同构表示附加伽罗瓦表示的理论很大程度上是基于PEL型的Shimura变体附加于幺正相似群上同调的研究。跟踪相似因子的需要使表示法变得复杂，但对最终结果没有影响。更自然的做法是使用与酉群本身相连的志村变种，它不会引入这些不必要的复杂性；然而，这些都是阿贝尔型，而不是PEL型，它们的上同调的伽罗瓦表示与更熟悉的志村变种的伽罗瓦表示略有不同。通过对PEL型Shimura变量的研究，建立了这些伽罗瓦表示的el函数的临界值。这些变体的自同构周期与那些附属于酉群的自同构周期是相同的，这一点并不明显，后者在相对示量公式的应用中更自然地出现，例如改进的Gan-Gross-Prasad猜想（Ichino-Ikeda和N. Harris的猜想）。本文利用附在酉群上的Shimura变量重新考虑了这些临界值，并得到了更简单的应用结果。
The theory of Galois representations attached to automorphic representations ofGL(n) is largely based on the study of the cohomology of Shimura varieties of PEL type attached to unitary similitude groups. The need to keep track of the similitude factor complicates notation while making no difference to the final result. It is more natural to work with Shimura varieties attached to the unitary groups themselves, which do not introduce these unnecessary complications; however, these are of abelian type, not of PEL type, and the Galois representations on their cohomology differ slightly from those obtained from the more familiar Shimura varieties.Results on the critical values of theL-functions of these Galois representations have been established by studying the PEL type Shimura varieties. It is not immediately obvious that the automorphic periods for these varieties are the same as for those attached to unitary groups, which appear more naturally in applications of relative trace formulas, such as the refined Gan-Gross-Prasad conjecture (conjecture of Ichino-Ikeda and N. Harris). The present article reconsiders these critical values, using the Shimura varieties attached to unitary groups, and obtains results that can be used more simply in applications.