Combining the CLAUDS & HSC-SSP surveys: U+grizy(+YJHKs) photometry and photometric redshifts for 18M galaxies in the 20 deg2 of the HSC-SSP Deep and ultraDeep fields

We present the combination of the Canada-France-Hawaii Telescope (CHFT) Large Area $U$-bands Deep Survey (CLAUDS) and the Hyper-Suprime-Cam (HSC) Subaru Strategic Program (HSC-SSP) data over their four deep fields. We provide photometric catalogs for $u$, $u^*$ (CFHT--MegaCam), $g$, $r$, $i$, $z$, and $y$ (Subaru--HSC) bands over $\sim 20~{\rm deg}^2$, complemented in two fields by data from the Visible and Infrared Survey Telescope for Astronomy (VISTA) Deep Extragalactic Observations (VIDEO) survey and the UltraVISTA survey, thus extending the wavelength coverage toward near-infrared with VIRCAM $Y$, $J$, $H$, and $K_s$ observations over $5.5~{\rm deg}^2$. The extraction of the photometry was performed with two different softwares: the HSC pipeline hscPipe and the standard and robust SExtractor software. Photometric redshifts were computed with template-fitting methods using the new Phosphoros code for the hscPipe photometry and the well-known Le Phare code for the SExtractor photometry. The products of these methods were compared with each other in detail. We assessed their quality using the large spectroscopic sample available in those regions, together with photometry and photometric redshifts from COSMOS2020, the latest version of the Cosmic Evolution Survey catalogs. We find that both photometric data sets are in good agreement in $Ugrizy$ down to magnitude$\sim26$, and to magnitude$\sim24.5$ in the $YJHK_s$ bands. We achieve good performance for the photometric redshifts, reaching precisions of $\sigma_{NMAD} \lesssim 0.04$ down to ${m}_i\sim25$, even using only the CLAUDS and HSC bands. At the same magnitude limit, we measured an outlier fraction of $\eta \lesssim 10\%$ when using the $Ugrizy$ bands, and down to $\eta \lesssim 6\%$ when considering near-infrared data. [abridged]