The Supreme Court gets to choose which cases it wants to hear, and it doesn't choose very many!!
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First of all court need some proof of your case, so that court should take some action.
send the case back to the lower court
supreme court of Missouri
A plaintiff or defendant in a federal court case (or in a state court case where a Federal Constitutional issue is in dispute ) who wants to appeal to the Supreme Court of the United States may ask for a writ of certiorari. The U.S. Supreme Court is obligated to take certain cases on appeal (for example, capital murder cases) but has discretion to take or not take certain others. The writ of certiorari is the Supreme Court's written agreement to take one of those discretionary cases on appeal.
The case Stone v. Graham took about three years to reach the United States Supreme Court. The initial ruling occurred in 1979 when the Kentucky Supreme Court decided the case, and it was subsequently appealed to the U.S. Supreme Court, which granted certiorari in 1980. The Supreme Court ultimately issued its decision in 1980.
send the case back to the lower court
No. The US Supreme Court has full discretion over cases heard under both its original and appellate jurisdiction, but is required to consider every case petitioned.
The ruling of the court below the Supreme Court will be upheld. The Supreme Court is similar to an appeals court. If they don't want to take the case, then whatever the court ruling was will stand.
There is no statutory time limit within which a decision must be rendered. Each case is unique and the length of time to consider it and render an opinion may depend much upon the importance, and complexity of the case under consideration.
The time it takes for a Court of Appeals to act on a case remanded by a state supreme court can vary significantly based on several factors, including the complexity of the case, the court's caseload, and procedural requirements. Generally, it can take anywhere from a few months to over a year. The court will typically prioritize the remanded case, but specific timelines can differ widely between jurisdictions.
The state supreme court, then the national supreme court. However, the defense must have a valid reason that the court should take the case, because both Supreme courts take far fewer cases than Criminal and Appeals courts do.