I am under the impression that online filing of patent applications is paper-free, i.e., I am not required to send any physical document to the USPTO, but sending all documents as PDF files is OK.
However, the USPTO is very precise about the oath: it has to be signed in presence of a notary, and the notary has to rubber-stamp it (at least in some states). Then, when filing online a patent application, is it enough to send a scanned copy of the signed and rubber-stamped oath as a PDF file? Maybe I also need to send the oath by snail mail to the USPTO? I have found no conclusive evidence.
There is also a substitute called a "declaration", but I have not found an officially sanctioned form for that in the USPTO website. The regulations sometimes seem to discourage it, sometimes the other way around.