There are numerous synonyms for the phrase "asserted under oath", each of which conveys a similar connotation of legal affirmation and truthfulness. Such synonyms might include "sworn", "attested", "declared under oath", "affirmed", "avowed", and "testified". These terms all suggest that the speaker is attesting to the veracity of their statements, and that their words are to be taken as reliable and binding in a legal context. Other possible synonyms include "validated", "certified", "endorsed", and "verified", each of which suggests that the statement has undergone some form of authoritative scrutiny or approval. Overall, these synonyms serve to underscore the seriousness and importance of statements made under oath in a legal setting.