Unless you can show that the drains were blocked before you rented the place then the landlord may be correct. Unblocking drains is normally a tenant responsibility. It can normally be done by buying and using a drain cleaner liquid from the supermarket. More persistent problems may need an acid cleaner such as One-Shot or rodding with a flexible tube. In the unlikely event that it does require the removal of pipes then that is something the landlord would have to do.
If you think this is caused by the flat below then try to persuade them to deal with it or go halves.