I agree with my colleagues. The position taken by your ex is the IRS's "default" - meaning in the absence of any agreement to the contrary, this defines who gets to take the relevant deduction. In New Jersey settlement agreements almost universally allocate the deductions between divorced parents to allow them both to benefit as much as possible, and they are routinely enforced.
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The following was taken from IRS Pub 4449.
Generally, because of the residency test, a child of divorced or separated parents is the qualifying child of the custodial parent. However, the child will be treated as the qualifying child (for the purposes of claiming a dependency exemption and the child tax credit, but not for the earned income credit) of the noncustodial parent if all four of the following statements are true:
a. are divorced or legally separated under a decree of divorce or separate maintenance,
b. are separated under a written separation agreement, or
c. lived apart at all times during the last 6 months of the year, whether or not they are or were married.
2. The child received over half of his or her support for the year from the parents.
3. The child is in the custody of one or both parents for more than half of the year.
4. The noncustodial parent attaches a Form 8332, or similar statement containing the same information required by the form, to his or her return. The form must be signed by the custodial parent. (See special rules in Publication 17 for a pre-1985 or post-1984 and pre-2009 divorce decree or separation agreement.)
See Publication 17 for additional rules for claiming an exemption for a dependent
It seems the custodial parent is confused. It sounds like everything under the bridge is copacetic.
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Mr. Walker has given you a superb answer. You should also check IRS Publication 501 at http://www.irs.gov/pub/irs-pdf/p501.pdf. Do not lose sleep over your ex reporting you to the IRS. The IRS has its hands full and does not have an interest in following up on tips like this. The IRS is only interested in making sure that only one parent claims the exemption, and that the parent who claims the exemption is actually entitled to it. The statute of limitations is three years from the date the return was filed. If you two had an agreement, then the court will uphold your agreement: but how are you going to prove your agreement in the absence of a writing. You need to have a written agreement with your ex. Yes, if she violates a court order, the judge MIGHT find her in contempt. But judges don't like to hold people in contempt.
Your ex spouse should follow the agreement in which you stated that you and your ex spouse modified the law to fit your case.
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