There appear to be some serious communications issues between you and your child's father. My sense is that your child's father may be planning to challenge your custody arrangements. I can only recommend that you contact a family law attorney as soon as possible to get this matter cleared up. What your child's father may not appreciate is that if the child is found to have been given an illegal or suspect admissions to a competitive entry school program (i.e. a lottery program) then the child may be forced to leave the school. Depending on the child's age this could be very upsetting and one has to question the judgment of an adult who does not anticipate the emotional harm that his/her conduct may cause a child.
This is not a substitute for a consultation with a family law attorney in your community.