Memoirs of the American Mathematical Society 1992; 142 pp; softcover Volume: 99 ISBN10: 0821825399 ISBN13: 9780821825396 List Price: US$32 Individual Members: US$19.20 Institutional Members: US$25.60 Order Code: MEMO/99/476
 Langlands theory predicts deep relationships between representations of different reductive groups over a local or global field. The trace formula attempts to reduce many such relationships to problems concerning conjugacy classes and integrals over conjugacy classes (orbital integrals) on \(p\)adic groups. It is possible to reformulate these problems as ones in algebraic geometry by associating a variety \(Y\) to each reductive group. Using methods of Igusa, the geometrical properties of the variety give detailed information about the asymptotic behavior of integrals over conjugacy classes. This monograph constructs the variety \(Y\) and describes its geometry. As an application, the author uses the variety to give formulas for the leading terms (regular and subregular germs) in the asymptotic expansion of orbital integrals over \(p\)adic fields. The final chapter shows how the properties of the variety may be used to confirm some predictions of Langlands theory on orbital integrals, Shalika germs, and endoscopy. Readership Researchers in the fields of automorphic forms, representation theory and \(p\)adic groups. Table of Contents  Basic constructions
 Coordinates and coordinate relations
 Groups of rank two
 The subregular spurious divisor
 The subregular fundamental divisor
 Rationality and characters
 Applications to endoscopic groups
