I found out today that in order to run Picasa on Fedora 11, you need to have the openssl-devel.i586 package installed. This package includes libssl.so, which is what Picasa needs when connecting to the Google site.
I found this odd, because -devel packages are normally not needed on non-developer machine. But just thought I would share this information.
You might also want to install nss-mdns.i586