Regular Trace Formula and Base Change Lifting

Abstract

Introduction. The Selberg trace formula has become a major tool in the study of automorphic forms on reductive groups. Although its underlying principle, of computing traces of representations by means of orbital integrals, is very simple, the standard expressions for this important formula are rather complicated; this makes applications hard to accomplish. The complexity of the expression for the formula may be due to the choice of truncation made in its proof. It would be advantageous to have a simple expression for the formula, at least for a set of test functions which is sufficiently large for applications. The possibility of its existence was suggested to us by some of Kazhdan's striking work on the trace formula (see, e.g., the density theorem of [Kl; Appendix], or the study of lifting in [K2]). Here we derive an asymptotic expression of this nature, in the simplest case of GL(2). For test functions with a component which is sufficiently with respect to all other components we obtain a simple, practical form of the trace formula. More precisely, if FU is a nonarchimedean local field and m 2 1 is an integer, we say that a locally constant function f, on Gu = GL(2, FU), which is supported on a compact-mod-center, is m-regular if it vanishes outside the open closed subset Sm = {zg-( ?)g; g in Gu; a, z in F,x with I a = 1 } (r denotes a uniformizer in F, x) of Gu, and its normalized orbital integral F(g ,fu) = A(g)4(g ,fu) is the characteristic function of Sm in G14. If F is a global field, u is a nonarchimedean place of F, and fu = ?, f, is a product over all places v ? u of F of smooth compactly supported mod-center functionsfv on Gv, such thatfv is the unit elementf v? of the Hecke algebra for almost all v, then we show that there exists mr0 = mo(fu) such that for any m 2r mo and m-regular functionfu = f (m), the regular test functionf = f (m) ? fu vanishes on the conjugacy classes in