You might have noticed we changed the SSL certificate for the api.twitter.com domain. If you're getting errors, you probably just need to install the new Root CA Certificate. You can obtain it directly from Verisign, or from this direct link: http://curl.haxx.se/ca/cacert.pem
i:/C=US/O=VeriSign, Inc./OU=Class 3 Public Primary Certification Authority - G2/OU=(c) 1998 VeriSign, Inc. - For authorized use only/OU=VeriSign Trust Network
For your Java Applications, be sure to have it installed in the Java Keychain, and think to restart your application. Note we also switched the IP to dedicated VIPs, so clearing your DNS Cache might also be necessary.
Update: If you need to get a list of PEM encoded certificates (including the Verisign one), we actually recommend the usage of Adam Langley's
extract-nss-root-certs tool, rather than downloading the cacert.pem file from haxx.se. Take a look on https://github.com/agl/extract-nss-root-certs for more info.