Memoirs of the American Mathematical Society 1999; 112 pp; softcover Volume: 137 ISBN10: 0821809598 ISBN13: 9780821809594 List Price: US$50 Individual Members: US$30 Institutional Members: US$40 Order Code: MEMO/137/655
 The trace formula is the most powerful tool currently available to establish liftings of automorphic forms, as predicted by Langlands principle of functionality. The geometric part of the trace formula consists of orbital integrals, and the lifting is based on the fundamental lemma. The latter is an identity of the relevant orbital integrals for the unit elements of the Hecke algebras. This volume concerns a proof of the fundamental lemma in the classically most interesting case of Siegel modular forms, namely the symplectic group \(Sp(2)\). These orbital integrals are compared with those on \(GL(4)\), twisted by the transpose inverse involution. The technique of proof is elementary. Compact elements are decomposed into their absolutely semisimple and topologically unipotent parts also in the twisted case; a double coset decomposition of the form \(H\backslash G/K\)where H is a subgroup containing the centralizerplays a key role. Readership Graduate students and research mathematicians working in automorphic forms, trace formula, orbital integrals, conjugacy classes of rational elements in a classical group and in stable conjugacy. Table of Contents Part I. Preparations  Statement of Theorem
 Stable conjugacy
 Explicit representatives
 Stable \(\theta\)conjugacy
 Useful facts
 Endoscopic groups
 Instability
 Kazhdan's decomposition
 Decompositions for \(GL(2)\)
 Decomposition for \(Sp(2)\)
Part II. Main comparison  Strategy
 Twisted orbital integrals of type (I)
 Orbital integrals of type (I)
 Comparison in stable case (I), \(E\slash F\) unramified
 Comparison in stable case (I), \(E\slash F\) ramified
 Endoscopy for \(H=GSp(2)\) type (I); Unstable twisted case. Twisted endoscopic group of type I.F.2; Twisted endoscopic group of type I.F.3, \(E\slash F\) unramified
 Twisted orbital integrals of type (II)
 Orbital integrals of type (II)
 Comparison in case (II), \(E\slash E_3\) ramified \((e=2)\); Unstable twisted case. Twisted endoscopic group of type I.F.2
 Comparison in case (II), \(E\slash E_3\) unramified \((e=1)\); Unstable twisted case. Twisted endoscopic group of type I.F.3
 Endoscopy for \(GSp(2)\), type (II)
 Comparison in case (III); Unstable twisted case. Twisted endoscopic group of type I.F.2
 Comparison in case (IV); Unstable twisted case. Twisted endoscopic group of type I.F.2
Part III. Semi simple reduction  Review
 Case of torus of type (I)
 Case of torus of type (II)
 Case of torus of type (III)
 Case of torus of type (IV)
 References
