On the Kuznetsov trace formula for PGL2(C)

Abstract

In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group Γ in PGL 2 pCq that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for PGL 2 pCq. This approach avoids the difficult 2010 Mathematics Subject Classification. 11F72, 11M36.

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

“…Unlike this, we start with one particular Poincaré series on G(A) and calculate its Whittaker-Fourier coefficient in two ways: by the spectral expansion in terms of automorphic functions, and by the geometric expansion in terms of orbital integrals associated with (Z(Q)B 0 (Q), U(Q))double cosets in G(Q). In this methodological aspect, our argument has similarity to [55]. Another remarkable feature of our formula is its simplicity, which is brought by the compact support condition of the test function f = ⊗ v f v .…”

A Whittaker-Plancherel Inversion Formula for SL2(ℂ)

“…This Bessel inversion formula is the bridge to the "Kloosterman form" of the Kuznetsov trace formula for SL 2 (C) from its "spectral form" in [BM5,LG]. There is however a direct approach to the Kloosterman form by a representation theoretic method of Cogdell and Piatetski-Shapiro in [Qi2]. Compare also [Kuz] and [CPS] for SL 2 (R).…”

A Voronoï--Oppenheim summation formula for number fields

“…The kernel formula for unitary representations of PGL 2 pRq first appears in the book of Cogdell and Piatetski-Shapiro [CPS], and has been generalized to GL 2 pRq and GL 2 pCq in [Qi5,§ §17,18]. For its applications to establishing the Kuznetsov formula and the Waldspurger formula, we refer the reader to [CPS,Qi1,BM1,CQ2,BM2,CQ1]. For the unitary case, the kernel formula is actually valid for all W φ papxqq in the Kirillov model.…”