Eisenstein Series and the Selberg Trace Formula. II

Abstract

The integral of the kernel of the trace formula against an Eisenstein series is investigated.The analytic properties of this integral imply the divisibility of the convolution L-function attached to a form by the zeta function of the field.Introduction.This paper is a sequel and generalization of [12], but can be read independently of that paper; in particular, we will repeat the description of the problem given in the introduction of [12], now, however, in an adelic setting.Let F be a global field, A its ring of adeles, and p0 the representation of PGL(2, A) by right translation on the space of cusp forms Lq(PGL(2, F)\ PGL(2,A)).Given any <p e C00O(PGL(2, A)), the operator p0(y) on this space is of Hilbert-Schmidt type and can be represented by a kernel function K0(x, y) for which an explicit formula of the form K0(x, y) = K(x, y) -Kms(x, y) -Ksp(x, v) is known, where K is given as a sum over PGL(2, F), KKs as an integral involving Eisenstein series, and Ksp as a sum of products of characters.In particular, one can calculate trp0(<p) from the identity trp0(<p) = / K0(x,x)dx;•,PGL(2,f)\PGL(2,A)the result is the Selberg trace formula.What we will do is to calculate instead the integral l(s) = I K0(x,x)E(x,s) dx,•/PGL(2, F)\PGL(2,A)where E(x, s) is an Eisenstein series.Our main result is an identity expressing I(s), roughly speaking, as a finite linear combination of zeta functions of quadratic extensions of F. Since the residue of E(x,s) at s = 1 is a constant function of x, one can in principle recover the Selberg trace formula from this identity by computing the residue of I(s) at s = 1, but the formula for I(s) has other interesting consequences.Most notably, it implies that, as in the special cases treated in [11 and 12], I(s) is divisible by £F(s) or, in other words, that the function K0(x, x) for any <p is orthogonal to the functions E(x, p) for all zeros p of the zeta function of F. A somewhat more precise formulation of the main result is as follows.

Locations

• Transactions of the American Mathematical Society - View - PDF