What is the ruling on telling my mother that I have vowed to fast for ten years, even though I have not made such a vow, so that she will stop asking me to break my fast?

Fasting one day and breaking fast one day is the dearest form of fasting to Allah, provided that the body is not weakened from fulfilling rights and obligations. In this case, obedience to the mother would have necessitated abandoning this fast, as it caused weakness. As for your telling your mother that you vowed to fast, if you did not truly vow, then it is a lie. Scholars have differed on the permissibility of lying in such situations for the sake of benefit; some have permitted it absolutely, while others have stipulated the use of equivocation and subtle hints. If the mother has ceased her prohibition, then there is no harm in continuing. However, if she returns and asks you to stop, then you must obey her. But if you truly vowed to fast, then you must fulfill it, and your mother has no right to prevent you from it, because obedience to parents is not obligatory when it comes to abandoning duties or vows.