Transfers of some Hecke elements for possibly ramified base change in GLn

In this paper, we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residual characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy–Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semisimple descent in the twisted space according to Waldspurger’s fundamental work, “L’endoscopie tordue n’est pas si tordue.”