  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 residue 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
semi-simple descend in the twisted space according to Waldspurger's fundamental
work "L'endoscopie tordue n'est pas si tordue".
