Lectures on the Arthur-Selberg trace formula1996

The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy classes of a group, and the spectral terms given by the induced representations. In general, these terms require a truncation in order to converge which leads to an equality of truncated kernels. The formulas are difficult in general and even the case of GL(2) is nontrivial. The book gives proof of Arthur's trace formula of the 1970s and 1980s with special attention given to GL(2). The problem is that when the truncated terms converge, they are also shown to be polynomial in the truncation variable and expressed as "weighted" orbital and "weighted" characters. In some important cases the trace formula takes on a simple form over G. The author gives some examples of this, and also some examples of Jacquet's relative trace formula. . This work offers for the first time a simultaneous treatment of a general group with the case of GL(2). It also treats the trace formula with the example of Jacquet's relative formula.