I have another idea. Nails should never be used in the construction of stairs, the nails will lose their grip and sing when they go up and down in the hole, like jb said.
Another way since your soo lucky to have access from below is to get 4 each 90degree metal angle supports or L brackets or shelve brackets for each tread. Get the ones that are about four to five inches long on each angle with three holes pre drilled in each side.
Get some heavy duty pan head screws enoough for each bracket.
Instructions for each tread, one squeeze some all purpose liquid nails under each tread and stringer, do one step at a time. then have a heavy set person go on the step that you just glued and stand on it, this will press the thread down.
Now set the angle brackets in place and screw them in forcing the tread down to the stringer. Do the other three on this tread. With four per thread this will hold it down.
Then procede to the next tread and do the exact same.
This should work,
But, the best way is to remove the carpet and just screw down each tread to that stringer.