How do I remove the shutdown button only on the login screen?

I don't want anyone to be able to shut down my computer from the login screen. Can I remove the shutdown button at the top right corner of the login screen in Ubuntu 12.04?

Even if you cannot hide the buttons, you can configure your system so that the lightdm user does not have permissions to shutdown or restart the system. Since the greeter runs as the lightdm user, this should solve your issue. Create two new files in /var/lib/polkit-1/localauthority/50-local.d, call them disable-lightdm-shutdown.pkla and disable-lightdm-restart.pkla and apply the contents as below. Note: You can also stick these into one file.

Shutdown:

[Disable Shutdown]
Identity=unix-user:lightdm
Action=org.freedesktop.consolekit.system.stop
ResultAny=no
ResultInactive=no
ResultActive=no

Restart:

[Disable Restart]
Identity=unix-user:lightdm
Action=org.freedesktop.consolekit.system.restart
ResultAny=no
ResultInactive=no
ResultActive=no

Go to /etc/lightdm/lightdm-gtk-greeter.conf and edit it to contain

indicators=false

This will remove all indicator from login page.