The FBI has authority to investigate specific crimes, and they fall within two categories: (1) National Security Priorities, which include counterterrorism, counterintelligence, and cyber crimes; and (2) Criminal Priorities, which include public corruption, civil rights, organized crime, white-collar crime, and
major theft/violent crime. If you go to the FBIs web site, you can click on links to get more specific information on each of these sub categories. Here is the link:
http://www.fbi.gov/hq.htm
the link says serial killers are included in the violent crimes, also interesting it states when the local LE are either not doing their jobs or hindering an investigation.