The determinative factor in terms of jurisdiction isn’t who arrested you on a child pornography charge since federal authorities intervene in any case where sexually explicit materials depicting a child are transported between states. More specifically, Federal jurisdiction is triggered whenever child pornography touches upon interstate or foreign commerce.
For example, if an individual in New Jersey meets someone in a chat room and downloads a file that originated from that person in California then a federal violation has been committed and the United States Attorney may exercise jurisdiction. Another way for a federal child pornography offense to arise is storing a file containing prohibited materials on a device or facility in another state.
Please note that the potential for federal authorities exercising jurisdiction is almost limitless whenever the internet is involved. Even if the child pornography image did not travel by mail or the like to another state or country, federal law jurisdiction arises if the computer used to store the image, where it originated or previously traveled in interstate or foreign commerce.
