To the IRS, the custodial parent is the parent with whom the child lived for the longer period of time during the year. With regard to your federal tax return, that is the only "custody" definition that matters.
Dependents 3 | Internal Revenue Service (irs.gov)
If you are the custodial parent by IRS definition, you may need to talk to a lawyer to see if the mother can be compelled to divulge the children's SSN's.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
