We have investigated the characteristics of solar wind and magnetospheric conditions associated with the occurrence of geosynchronous relativistic electron events. Most of the geosynchronous relativistic events for April 1999 to December 2002 are found to occur during prolonged (a number of days) quiet intervals following the appearance of high-speed solar wind streams. In a typical relativistic event, the electron fluxes begin to increase by orders of magnitude when the solar wind density drops after reaching a sharp peak at the leading edge of a high-speed stream. The increased fluxes stay at a high level until the quiet solar wind conditions cease. In addition, enhanced ULF wave activity and substorm injections of 10s to 100s keV electrons are observed at the time of the large flux increases in the events. We found that geosynchronous relativistic events can be observed only when both the solar wind and magnetospheric wave/substorm injection conditions are favorable regardless of whether or not a magnetic storm takes place. These observations suggest the following scenario for the occurrence of a geosynchronous relativistic electron event: (1) Quiet solar wind conditions (i.e., no strong solar wind pressure and large southward turnings of IMF Bz) can lead to stable and more dipole-like magnetospheric configurations in which the geosynchronous orbit is located well inside the trapping boundary of the energetic electrons. (2) If a large population of MeV electrons is generated (by some acceleration process(es) involving enhanced ULF wave and substorm injections) in the inner magnetosphere, it can be trapped and effectively accumulated to a high intensity. (3) The high electron flux can persist for a number of days in the geosynchronous region as long as the solar wind conditions remain quiescent. The occurrence of a geosynchronous relativistic electron event requires not only the proper acceleration process and sufficient seed electrons but also no significant loss process that dominates over any acceleration/source.