OK, after putting in the magic words to Google (I tried Open Sesame and everything) it revealed this previous StackOverflow question: Asp.Net Forms Authentication SSO between 4.0 and 2.0 applications
This eventually did the trick. I say eventually because I had modified a few of the web.config settings for both my login application as well as my new 4.0 app and it took me a bit to discover that I had added Decrytion="3DES" to the v4 app but no longer had that in my login app. Ah, .net, what a tangled wwweb we weave...