I have done some independent research that I wish to mention in my SOP. The full account of what I have done till has been published as an e-print on arxiv.org. Can I give a hyperlink to this e-print in my SOP?
I would differ a bit from both of the previous two responses.
The SOP and the CV serve different purposes. The SOP is the high-level summary (some would say sales pitch) for why your work is interesting and why you should be hired. This needs to be tightly written and compelling, and part of being compelling is highlighting particularly unique, interesting, or high-profile publications. Importantly, I've been told by several hiring committee members that more people will read your SOP than your CV - so if something's important, don't bury it only in the CV. The CV is more of a "data dump" in which you list everything that you've done that may be of interest. You don't necessarily want all of this detail clouding up your SOP, although you want the hiring committee to have access to it if necessary. Think of your SOP like a glossy brochure and the CV like a dictionary - the former is intended to be read while the latter is intended to be referenced.
So, that said, I think it's both a mistake to leave your publication out of your SOP entirely and also to list all of your publications there (if you have many). Reference only the key ones that support the rest of your story and include a hyperlink in a footnote if you think that a reader might actually follow it. I would suspect that this will be a rare event, though, and I would encourage you to put enough detail from this publication in the SOP itself that a reader won't have to follow the link to get the gist of what you did.
Yes, you should absolutely list it in your statement, in the bibliography, with a hyperlink, along with all the other papers you cite in your statement.
No, you shouldn't list it on your Statement of Purpose. You should cite it properly in your CV / Resume, or in a separate list of publications.
You can certainly discuss the research that led to the paper in your statement, but if the statement-readers would like to read the paper itself, they can find the reference in your other material.