What does the notice from the IRS say? What happens next is that you respond to the notice. If it is asking for information or documents, you send them what they are asking for.
The tax law doesn't care what the court agreement says. The court can't override the tax law. Under the tax law, the custodial parent is the one that the child lived with for more nights during the year. That's all that matters. Only the custodial parent can claim the child as a dependent, unless the custodial parent gave the noncustodial parent a signed Form 8332 allowing the noncustodial parent to claim the child.
If the custodial parent releases the exemption to the noncustodial parent by signing Form 8332, there is a rule governing how the tax benefits associated with the child are split between the two parents. The noncustodial parent can claim the child as a dependent, and claim the child tax credit and an education credit if applicable. The custodial parent can still use the child for the Earned Income Credit (EIC), filing as head of household, and the child care credit.
If you got the IRS notice because you and your ex both claimed the same child, then your ex also got a notice. After you both respond to the notice, the IRS will determine which one of you is allowed to claim the child as a dependent. The one who is not allowed to claim her will have to pay additional tax plus interest and penalties.
