Magnetic reconnection at the subsolar magnetopause is persistently accompanied by strong fluctuations of the magnetic field (B), plasma density (n), and all components of the electric field (E) and current (J). The strongest fluctuations are at frequencies below the lower hybrid frequency and observed in a thin, intense current sheet adjacent to the electron diffusion region. In this current sheet, the background magnitudes of B and n are changing considerably, creating an inhomogeneous plasma environment. We show that the fluctuations in B and n are consistent with an oscillatory displacement of the current sheet in the surface normal direction. The displacement is propagating parallel to the magnetic reconnection X line. Wavelengths are on the order of or longer than the thickness of the current sheet to which they are confined, so we label these waves electromagnetic drift waves. E and J fluctuations are more complex than a simple displacement. They have significant variations in the component along B, which suggest that the drift waves also may be confined along B. The current sheet is supported by an electron drift driven by normal electric field, which, in turn, is balanced by an ion pressure gradient. We suggest that wave growth comes from an instability related to the drift between the electrons and ions. We discuss the possibility that drift waves can displace or penetrate into the electron diffusion region giving magnetic reconnection three-dimensional structure. Drift waves may corrugate the X line, possibly breaking the X line and generating turbulence.