General Discussion

Trump's lawyers will file a petition for a writ of certiorari, asking the Court to take the appeal. The petition will contain arguments and case law supporting it's request, but there are no briefs and no hearing (oral argument). The Court will decide based on that whether to hear the appeal or to let the lower court ruling stand.

Even if the Court decides to take the case, that's no indication of how they'll eventually rule or cause to be concerned that they'll overturn the appellate court ruling that Trump turn over his taxes. In fact, it's possible the Court could take the case in order to shut Trump's immunity arguments down across the board once and for all.

As I wrote a few weeks ago, the appellate court opinion was so carefully and narrowly drafted, the judge left very little for Trump to appeal. And the only issue they can appeal is so clear-cut, it would be difficult for the Supreme Court to find any fault with it. That doesn't mean it won't reverse it or go beyond the narrow ruling - the Court can pretty much do what it wants - but it's unlikely.