Deep GMRT 610 MHz Observations of the ELAIS N1 Field : Catalogue and
  Source Counts
  This is the first of a series of papers based on sensitive 610 MHz
observations of the ELAIS N1 field using the Giant Metrewave Radio Telescope.
We describe the observations, processing and source catalogue extraction from a
deep image with area of 1.86\,deg$^{2}$ and minimum noise of
$\sim$7.1\,$\mu$Jy/beam. We compile a catalogue of 4290 sources with flux
densities in the range 28.9\,$\mu$Jy - 0.503\,Jy, and derive the
Euclidean-normalized differential source counts for sources with flux densities
brighter than $\rm{35.5\,\mu\,Jy}$. Our counts show a flattening at 610 MHz
flux densities below 1 mJy. Below the break the counts are higher than previous
observations at this frequency, but generally consistent with recent models of
the low-frequency source population. The radio catalogue is cross-matched
against multi-wavelength data leading to identifications for 92\% and reliable
redshifts for 72\% of our sample, with 19\% of the redshifts based on
spectroscopy. For the sources with redshifts we use radio and X-ray luminosity,
optical spectroscopy and mid-infrared colours to search for evidence of the
presence of an Active Galactic Nucleus (AGN). We compare our identifications to
predictions of the flux density distributions of star forming galaxies (SFGs)
and AGN, and find a good agreement assuming the majority of the sources without
redshifts are SFGs. We derive spectral index distributions for a sub-sample.
The majority of the sources are steep spectra, with a median spectral index
that steepens with frequency;
$\mathrm{\alpha^{325}_{610}\,=\,-0.80\,\pm\,0.29}$,
$\mathrm{\alpha^{610}_{1400}\,=\,-0.83\,\pm\,0.31}$ and
$\mathrm{\alpha^{610}_{5000}\,=\,-1.12\,\pm\,0.15}$.
