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The SRG/eROSITA All-Sky Survey. The Weak-Lensing Mass Calibration and the Stellar Mass-to-Halo Mass Relation from the Hyper Suprime-Cam Subaru Strategic Program

We present the weak-lensing mass calibration and constrain the BCG (brightest cluster galaxy) stellar-mass-to-halo-mass-and-redshift ($M_{\star,\mathrm{BCG}}-M-z$) relation for a sample of $124$ galaxy clusters and groups at redshift $0.1<z<0.8$ from the first Data Release of the $eROSITA$ All-Sky Survey (eRASS1), using data from the Hyper Suprime-Cam (HSC) Subaru Strategic Program. The cluster survey is conducted by the $eROSITA$ X-ray telescope aboard the Spectrum-Roentgen-Gamma (SRG) space observatory. The cluster sample is X-ray-selected and optically confirmed with a negligibly low contamination rate ($\approx5%$). On a basis of individual clusters, the shear profiles of $96$ clusters are derived using the HSC Three-Year (HSC-Y3) weak-lensing data, while the BCG stellar masses of $101$ clusters are estimated using the SED template fitting to the HSC five-band ($grizY$) photometry. The observed X-ray photon count rate is used as the mass proxy, based on which individual halo masses are obtained at the given count rate in a population modelling while accounting for systematic uncertainties in the weak-lensing modelling through a simulation-calibrated weak-lensing mass-to-halo-mass relation. The count rate and BCG stellar mass relations are simultaneously constrained in a forward and population modelling. In agreement with the results based on the weak-lensing data from the DES and KiDS surveys, we obtain a count rate relation with a self-similar redshift scaling and a mass trend that is steeper than the self-similar prediction. Our results suggest that the BCG stellar mass at a fixed halo mass has remained stable with a moderate increase at a level of $\left(20\pm8\right)%$ since redshift $z\approx0.8$. This finding supports the picture of the ``rapid-then-slow'' BCG formation, where the majority of the stellar mass must have been assembled at much earlier cosmic time.