For anyone to sign a birth certificate, a legal document, they are asserting that they are the legal parent of that child.
In the event that your husband has not adopted your son, I strongly advise that, because then during the adoption process, you may be able to have a new birth certificate issued (though that is something best discussed with an attorney specializing in adoptions). Also, adoption would provide your husband with full legal rights re: your son. Sufficient legal notice per the requirements of the Ga adoption statutes would have to be provided to the real father; notwithstanding, if there is good cause for formally severing all legal rights the father may have to your son (regardless of what he may or may not have done previously), that would be done during the adoption also. Good luck.Ask a similar question
Your husband would be committing a crime in making a knowingly false statement.
If he wants parental rights, see a lawyer about adoption. An adoption, once complete, will add him as the father to the birth certificate.
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