An approach to nonsolvable base change and descent (Q2906835)

The paper records some of the author's attempts to attack non-solvable base change of automorphic representations of \(\mathrm{GL}_n\) using trace identities, and gives evidences for its viability. Let \(E/F\) be a finite Galois extension of number fields. When \(E/F\) is cyclic of prime degree, the base change is established by \textit{J. Arthur} and \textit{L. Clozel} via twisted endoscopy [Simple algebras, base change, and the advanced theory of the trace formula. Princeton, NY: Princeton University Press (1989; Zbl 0682.10022)], and the solvable base change follows. On the other hand, the non-solvable case reduces easily to the case of simple non-abelian \(\mathrm{Gal}(E/F)\). The paper handles the latter aspect by using (i) comparison of trace formulas and (ii) Langlands' idea of ``beyond endoscopy''. Following ideas of Langlands, Sarnak et al., it makes use of carefully crafted test functions and certain limiting forms of the cuspidal spectrum.NEWLINENEWLINEUnder several technical assumptions that are too complicated to state here, one obtains a family of conjectural trace identities for establishing the base change. It is observed that the limiting forms can be rewritten in terms of orbital integrals using either the usual trace formula, or the Kuznetsov-Bruggeman relative trace formula, thereby giving some hope for the comparison.