Geometric aspects of the Trace Formula
Geometric aspects of the Trace Formula
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微量公式的几何方面
DOI：
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发表时间：
2016
期刊：
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通讯作者：
O. Buisan
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作者：
M. Arzoz;L. Ibarz;J. Fernández;Manuel Valiente;J. Roca;S. Edo;O. Buisan
Arthur-selberg痕迹公式是塞尔伯格在1950年代开发的现代工具，他用它来确定MAASS形式的存在，以实现SL的一致性亚组（2，r）。 。 G（a）上的两个分布，一个在光谱数据中定义的几何数据，是Arthur工作的关键问题。有力研究痕迹的案例像塞尔伯格的环境一样，该小组的公式可以研究给定的G型。频谱和几何方面的融合仍然存在一些重要的问题。及其应用程序。
The Arthur-Selberg trace formula is a central tool in the modern theory of automorphic forms. It was developed by Selberg in the 1950s where he used it to establish the existence of Maass forms with respect to congruence subgroups of SL(2,R). Later on Arthur, driven by Langlands’ functoriality conjectures, developed the trace formula for adelic quotientsG(F )G(A) of an arbitrary reductive group G over a number field F . The trace formula is an identity between two distributions on G(A), one defined in terms of spectral data and other one in terms of geometric data. A key issue in Arthur’s work is the comparison of trace formulas of two different groups. Besides functoriality, the trace formula (combined with the Lefschetz trace formula) admits striking applications to the Langlands correspondence (both global and local) and to the arithmetic of Shimura varieties. There is however, a strong case to study the trace formula on the group itself, much like as in Selberg’s context. One can use it to study the distribution of the automorphic spectrum of the given group G. Examples of results of this nature are the Weyl law, the limit multiplicity problem, and SatoTate equidistribution of families of automorphic forms. Applications to spectral theory lead to difficult analytic problems related to the trace formula. Some of these problems have been settled recently. This is the absolute convergence of the spectral and the geometric side. There are still some outstanding problems which are important for applications. An important variant of Arthur’s trace formula is Jacquet’s relative trace formula which is pertaining to (affine) symmetric spaces, or more generally, spherical varieties. It is intimately related to the analysis of period integrals and distinguished representations. Recently, there has been a lot of progress on the analysis of the relative trace formula and its applications. However, some foundational questions remain open.