The nature of the close magnetic white dwarf + 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 121209.31+013627.7 reveals pulse-like variability in all bands from i to u, 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 minutes. This period is commensurate with the radial velocity period determined by Schmidt et al. 2005 of ~90 minutes, and consistent with the rotation period of the accretor being equal to the binary orbital period. We combine our observations with those recently published by Koen and Maxted 2006 to provide an accurate ephemeris. We also detect the system in X-rays with Swift, and estimate the accretion rate at ~1x10^-13Msun per year. We suggest that SDSS1212 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. Six such post-common envelope, ``pre-Polar systems - termed ``low accretion rate Polars (LARPs) by Schwope et al. 2002 - have previously been identified through optical cyclotron emission lines. Cyclotron emission from SDSS1212 has recently been detected in the near-IR Debes et al. 2006 but, if detached, it would be the first ``LARP with a probably sub-stellar secondary. 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 over-flow and wind accretion scenarios.