The role of the U.S. Supreme Court is to ensure that the lower federal courts (and the state level courts) have correctly interpreted and applied constitutional and federal law.
The Constitution gives the Supreme Court authority to rule state or Federal Laws unconstitutional. It can also order acquittals or new trials on the basis of violations of the U.S. Constitution or federal statutes. The Court also has the authority to investigate any criminal cases that have loopholes, unanswered questions, anything that wasn't clear in the first ruling and if someone feels their constitutional rights have been violated.
It exercises final appellate jurisdiction in cases involving federal law and it has original jurisdiction in a limited number of matters. The Supreme Court hears only cases that present a substantial federal question. A case cannot be appealed to the Supreme Court until it has exhausted all other possible remedies, including lower court and any applicable state court appeals.
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US Supreme Court
The [State] Supreme Court (or its equivalent) has final authority unless the question being addressed in the state constitution conflicts with the US Constitution, in which case the US Supreme Court has final authority.
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The supreme Court was overwhelmed by cases.
Yes, of course it's legal to petition the US Supreme Court. When discussing Supreme Court cases, it's important to realize "petition" means to file an appeal of a case already decided by the lower courts. It does not mean to send a signature petition in protest or support of a decision. You are within your legal rights to send such a document, but the Court is not required to read it.