The birth certificate doesn't matter. But you only need a written agreement from the other parent if the child lives with the other parent more than half the year and not with you.
For a child under age 19, the only parent with the automatic right to claim a child as a dependent is the parent where the child lives more than half the nights of the year. If this is you, you can claim the child and do not need any forms or agreements involving the other parent. If the child lives with the other parent more than half the year, then you can't claim the child unless you have a signed release from the custodial parent. If the child is living with their actual parent, that is true regardless of the paperwork.
If the child lives with both parents (because you are unmarried and lived together for part or all of the year), then you don't need a signed agreement either. One parent claims the child and answers "no" to the question about a custody agreement, because that question only is asking about legal agreements between parents who live apart. The other parent leaves the child completely off their tax return, don't even mention the child.
If the child is not living with you, but the person they are living with is not their biological parent or parent by marriage (stepparent) then things get complicated and we would need more information.
