×

Rank-one residues of Eisenstein series. (English) Zbl 0711.11022

Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Pt. II: Papers in analysis, number theory and automorphic L-functions, Pap. Workshop L-Funct., Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 3, 111-125 (1990).
[For the entire collection see Zbl 0698.00021.]
In [Ann. Math., II. Ser. 130, No.3, 473-529 (1989; Zbl 0701.11019)] W. Müller has given a proof of the trace class conjecture. To state it, let G denote a real reductive group and \(\Gamma\) an arithmetic lattice in G. Any function f on G which is infinitely differentiable and of compact support acts by convolution on the space \(L^ 2(\Gamma \setminus G)\). Let \(L^ 2_{dis}\) denote the Hilbert sum of all simple G-submodules of \(L^ 2(\Gamma \setminus G)\). The trace class conjecture states that \(\cdot *f\) is of trace class on \(L^ 2_{dis}\). The proof proceeds in two steps. By the theory of Eisenstein series the space \(L^ 2_{dis}\) is decomposed into a sum of contributions attached to classes of parabolic subgroups. The first step is to solve the trace class problem for a maximal proper parabolic subgroup. Then W. Müller deduces the general assertion from this by means of a careful analysis of Eisenstein series. The method for the first step relies on classical techniques from the spectral theory of differential operators.
The paper under consideration offers an alternative approach to the first step. The basic idea is to use the convergence of certain terms in the trace formula as developed by J. Arthur. From this the author deduces estimates on the eigenvalues which are weaker than the ones given by Müller but suffice for the final result and are proven in a way more in the spirit of the rest of the proof.
Reviewer: A.Deitmar

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F11 Holomorphic modular forms of integral weight