Miller effect: if a capacitor (C) is placed between input and output of an inverting amlifier,the equivalent input capacitance becomes (1+gain)*C
When the transistor is in common emmiter mode,the base-collector capacitance is this miller capacitor.Since the gain is high,the input capacitance is high,too.It equals Cb-e+ (1+gain)*Cc-b
When the common base mode is used,the base-collector capacitor does not affect the input since it is grounded at the base end.The input capacitanse is equal to Cbe.(Well,in practise there is a small parasitic capacitance between collector and emmiter)
Common collector circuit has the collector end grounded (Vcc is ground for AC) , so the input capacitance equals the base-collector capacitance provided the load has no capacitance of its own.