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Write your name at the bottom of the form.
Literal
Form [possessive-の] bottom [at-に] name [object-を] write.
下部 ('bottom portion / lower part') is a formal compound: 下 ('below') + 部 ('section / part'). It's the slightly more formal/written counterpart to colloquial 一番下 ('the very bottom'), typical of clerical instructions on forms and documents.