In the development of the Beyond Endoscopy program to date, a lot of effort has focused on finding a geometric expression for the "Ramanujan'" spectrum of the Arthur--Selberg trace formula, i.e., removing the non-tempered Arthur packets. An early suggestion by Sarnak was to circumvent this problem altogether, by using the Kuznetsov formula instead, where only the Ramanujan spectrum appears.

I will report on ongoing joint work with Chen Wan, where we directly compare the two trace formulas for GL(n), including explicit calculations of the "transfer operators'' that isolate the non-Ramanujan spectrum in low rank. For GL(2), our work generalizes Rudnick's PhD thesis, and this operator coincides with the Fourier transform on the affine parameter space of orbital integrals, which has already appeared in works of Frenkel–Langlands–Ngô and Altuğ; but in higher rank it has a different form. Further motivation for this work comes from the Relative Langlands program, where there is a hope that non-standard comparisons of (relative) trace formulas could be used to prove conjectural relationships between periods of automorphic forms and special values of L-functions.