We present the first results from the largest deep extragalactic millimetre-wavelength survey undertaken to date. These results are derived from maps covering over 0.7 deg^2, made at 1.1mm, using the AzTEC continuum camera mounted on the James Clerk Maxwell Telescope. The maps were made in the two fields originally targeted at 0.85mm with SCUBA in the SHADES project, namely the Lockman Hole East (mapped to a depth of 0.9-1.3 mJy rms) and the Subaru XMM Deep Field (1.0-1.7 mJy rms). The wealth of existing and forthcoming deep multi-frequency data in these two fields will allow the bright mm source population revealed by these images to be explored in detail in subsequent papers. Here we present the maps themselves, a catalogue of 114 high-significance sub-millimetre galaxy detections, and a thorough statistical analysis leading to the most robust determination to date of the 1.1mm source number counts. Through careful comparison, we find that both the COSMOS and GOODS North fields, also imaged with AzTEC, contain an excess of mm sources over the new 1.1mm source-count baseline established here. In particular, our new AzTEC/SHADES results indicate that very luminous high-redshift dust enshrouded starbursts (S_{1.1} > 3 mJy) are 25-50% less common than would have been inferred from these smaller surveys, thus highlighting the potential roles of cosmic variance and clustering in such measurements. We compare number count predictions from recent models of the evolving mm/sub-mm source population to these SMG surveys, which provide important constraints for the ongoing refinement of semi-analytic and hydrodynamical models of galaxy formation, and find that all recent models over-predict the number of bright sub-millimetre galaxies found in this survey.
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