You can screw through it to the subfloor, but it is brittle stuff and will crack up when you do that. If you have a wood subfloor it would make more sense to use a 1/4" crown power stapler.
Personally, I would remove it. I can't advise you to do that, but I would keep it wet and scrape it up with a razor floor scraper. Since it is brittle, it will crack and lift off fairly easily.
HOWEVER, the adhesive is usually black stuff that also contains asbestos. You can get a trowel-on sealer to cover it and then go from there.