Title: The effect of longitudinal spin-fluctuations on high temperature properties of Co$_3$Mn$_2$Ge

Abstract: It is demonstrated that thermally induced longitudinal spin fluctuations (LSF) play an important role in itinerant Co$_3$Mn$_2$Ge at an elevated temperature. The effect of LSF is taken into account during {\it ab initio} calculations via a simple model for the corresponding entropy contribution. We show that the magnetic entropy leads to the appearance of a medium size local moment on Co atoms. As a consequence, this leads to a renormalization of the magnetic exchange interactions with a quite substantial impact upon the calculated Curie temperature. Taking LSF into account, the calculated Curie temperature can be brought to be in good agreement with the experimental value.