The environment of L1512, a starless dense core, has been mapped at high angular resolution in the 12CO (J=2-1) line over more than 1 pc, with a few positions observed in the 12CO (J=3-2) and (J=4-3) lines. The gas outside the dense core is structured in several filaments, roughly 1 pc long and ~0.1 pc thick, converging at the dense core position. Small longitudinal (~1 km s-1 pc-1) but large transverse (up to 8 km s-1 pc-1) velocity gradients are observed. Remarkably, the transverse gradients can be seen to change sign periodically, along at least one of the filaments. Thus, there are oscillations in the toroidal velocity within the filaments, which may be a signature of a magnetohydrodynamic instability developing in filaments permeated by a helical magnetic field. In the case of L1512, according to the analysis of Fiege & Pudritz, the growth rate of the instability is low, corresponding to a timescale of the order of 1 Myr. We deduce from the wavelength of the oscillations that the toroidal component of the magnetic field dominates the poloidal component. The toroidal component helps confine the filaments, which are not otherwise confined by self-gravity (m/mvir~0.2), by the pressure of the galactic H I layer, or by external turbulent pressure. We find that the velocity gradients in the vicinity of the dense core provide an estimate for an upper limit to the accretion rate onto the dense core of M=4×10-6 Msolar yr-1. For the gas characteristics in the filaments, we find that a broad range of density and temperature is allowed for the gas, from nH2=2×103 cm-3 for the coldest case (Tk=20 K) down to nH2=180 cm-3 for the warmest (Tk=250 K).