The nature of the close magnetic white dwarf plus probable brown dwarf binary SDSS J121209.31+013627.7

Optical time series photometry of the short-period magnetic white dwarf + probable brown dwarf binary SDSS J121209.31+013627.7 reveals pulse-like variability in all bands from i' to u', increasing towards bluer wavelengths and peaking at u'. These modulations are most likely due to a self-eclipsing accretion hot spot on the white dwarf, rotating into view every 88.43 min. This period is commensurate with the Ha radial velocity period of; 90 min, and consistent with the rotation period of the accretor being equal to the binary orbital period. We combine our observations with other recently reported results to provide an accurate ephemeris. We also detect the system in X-rays with Swift, and estimate the accretion rate at approximate to 10(-13) M-circle dot yr(-1). We suggest that SDSS J121209.31+013627.7 is most likely a magnetic cataclysmic variable in an extended state of very low accretion, similar to the well-studied polar EF Eri. Alternatively, the putative brown dwarf is not filling its Roche lobe and the system is a detached binary in which the white dwarf is efficiently accreting from the wind of the secondary. However, it is unclear whether an L dwarf wind is strong enough to provide the measured accretion rate. We suggest further observations to distinguish between the Roche lobe overflow and wind accretion scenarios.