Should he kill the pigeons that land on his roof and cause a nuisance?

If the pigeons belong to someone, then it is not permissible to slaughter them except with the permission of their owner. He can rid himself of this nuisance by scaring them away, and not giving them a place to land. 

If they do not belong to anyone, then he may slaughter them and make use of them. 

The Standing Committee for Issuing Fatwas was asked: In our house there are a lot of pigeons whose owner is not known, and they have increased in number, making a lot of dirt and causing us trouble. Is it acceptable to hunt them and eat them or keep them in cages? 

They replied: You can protect your houses against the birds entering by scaring them off, and not giving them room to settle there or feeding them. Thus you will be spared the nuisance and dirt. As for hunting them or keeping them in cages so that they become your property, that is not permissible except with the permission of their owner. End quote. 

Fataawa al-Lajnah al-Daa’imah (22/512) 

And Allaah knows best.