Is it permissible for me to return to my wife, whom I divorced by phone three times during a period of purity in which I had intercourse with her, after my brother admitted that the reason for the divorce—which was her slandering my father—was a lie?

Divorce pronounced during a period of purity (Tuhr) in which intercourse has occurred is an innovated (bid'i) and forbidden divorce, but it is considered valid by most scholars. Your statement to your wife, "You are divorced, you are divorced, you are divorced," effects a single divorce unless you intended three. If you have not divorced her more than once before, you can revoke the divorce as long as she is within her waiting period (iddah). If the waiting period ends, you can remarry her with a new contract. Some scholars hold the view that divorce does not occur at all if it was conditioned on something that later proved to be false, such as divorcing a wife due to suspicion of her conduct, and then her innocence is established. Therefore, it is advisable to present the matter to a Sharia court or to trustworthy scholars.