  The VISTA near infrared Y, J, Ks survey of the Magellanic System (VMC) is
collecting deep Ks band time series photometry of pulsating variable stars
hosted by the two Magellanic Clouds and their connecting Bridge. In this paper,
we present Y, J, Ks light curves for a sample of 4172 Small Magellanic Cloud
(SMC) Classical Cepheids (CCs). These data, complemented with literature V
values, allowed us to construct a variety of period-luminosity (PL),
period-luminosity-color (PLC), and period-wesenheit (PW) relationships, valid
for Fundamental (F), First Overtone (FO) and Second Overtone (SO) pulsators.
The relations involving V, J, Ks bands are in agreement with their counterparts
in the literature. As for the Y band, to our knowledge we present the first CC
PL, PW, and PLC relations ever derived using this filter. We also present the
first near infrared PL, PW, and PLC relations for SO pulsators to date. We used
PW(V,Ks) to estimate the relative SMC-LMC distance and, in turn, the absolute
distance to the SMC. For the former quantity we find a value of
Delta(mu)=0.55+-0.04 mag, in rather good agreement with other evaluations based
on CCs, but significantly larger than the results obtained from older
population II distance indicators. This discrepancy might be due to the
different geometric distributions of young and old tracers in both Clouds. As
for the absolute distance to the SMC, our best estimates are
mu(SMC)=19.01+-0.05 mag and mu(SMC)=19.04+-0.06 mag, based on two distance
measurements to the LMC, which rely on accurate CC and eclipsing Cepheid binary
data, respectively.
