If you make a written statement to the police or another recognized authority like a notary public that you swear is the truth, the whole truth, and nothing but the truth, what you've put down on paper is an affidavit.
We get the noun affidavit from the Latin affidare, which means "to pledge." When you state something in an affidavit, you're pledging to the powers that be that you're not lying. The word usually appears in legal settings; if you watch a lot of cop shows or courtroom dramas, it probably rings a bell. "According to the affidavit, Mrs. Smith told police she found the burglar in her closet wearing her most expensive pair of shoes."