  We present a current and charge conserving theory for the low frequency
admittance of a two-dimensional electron gas connected to ideal metallic
contacts and subject to a quantizing magnetic field. In the framework of the
edge-channel picture, we calculate the admittance up to first order with
respect to frequency. The transport coefficients in first order with respect to
frequency, which are called emittances, determine the charge emitted into a
contact of the sample or a gate in response to an oscillating voltage applied
to a contact of the sample or a nearby gate. The emittances depend on the
potential distribution inside the sample which is established in response to
the oscillation of the potential at a contact. We show that the emittances can
be related to the elements of an electro-chemical capacitance matrix which
describes a (fictitious) geometry in which each edge channel is coupled to its
own reservoir. The particular relation of the emittance matrix to this
electro-chemical capacitance matrix depends strongly on the topology of the
edge channels: We show that edge channels which connect different reservoirs
contribute with a negative capacitance to the emittance. For example, while the
emittance of a two-terminal Corbino disc is a capacitance, the emittance of a
two-terminal quantum Hall bar is a negative capacitance. The geometry of the
edge-channel arrangement in a many-terminal setup is reflected by symmetry
properties of the emittance matrix. We investigate the effect of voltage probes
and calculate the longitudinal and the Hall resistances of an ideal
four-terminal Hall bar for low frequencies.
