Is it permissible for someone who enters the mosque and finds the Imam performing the prayer, while he (the newcomer) is without ablution and there is no water in the mosque, to perform tayammum and pray, or should he go home to perform ablution, thus missing the virtue of congregational prayer, given that water is available in his home?

The virtue of performing congregational prayer is not among the reasons that permit tayammum (definition) (dry ablution). Therefore, a person in a state of minor ritual impurity (al-muḥdith) must go out to seek water. Tayammum is only permissible if one fears that the prayer time will expire due to seeking or using water. If he does not find water, or if the water is insufficient, then tayammum is permissible for him, based on the Almighty’s saying: ﴿And if you do not find water, then seek clean earth and wipe over your faces and hands with it.﴾