On or before the 10th day of August of each year following such final determination, the secretary shall file with the county auditor a copy of the report with a statement endorsed thereon over his signature that it has been finally adopted by the board and the auditor shall enter the amounts of the assessment against the respective lots or parcels of land as they appear on the current assessment roll. If a district collects taxes pursuant to Chapter 3 (commencing with Section 31730), Part 7 of this division, the secretary shall file the report with the collector.