Ezra 5:8 - 10

"The king should know that we went to the district of Judah, to the temple of the great God. The people are building it with large stones and placing the timbers in the walls. The work is being carried on with diligence and is making rapid progress under their direction. We questioned the elders and asked them, 'Who authorized you to rebuild this temple and restore this structure?' We also asked them their names, so that we could write down the names of their leaders for your information." NIV translation

Their report informed the king that the people were focused on their task and making rapid progress. (We must remember that they had been given a pretty strong message from God by the prophets.)

They were asked their names in hopes that the king would authorize the opposition to round them up and to stop the progress.

Ezra 5:11 & 12

"This is the answer they gave us: 'We are the servants of the God of heaven and earth, and we are rebuilding the temple that was built many years ago, one that a great king of Israel built and finished. But because our fathers angered the God of heaven, he handed them over to Nebuchadnezzar the Chaldean, king of Babylon, who destroyed this temple and deported the people to Babylon.'" NIV translation

Instead of giving them a list of names, the exiles had given them a short history lesson. They recalled the fact that Israel had once had a great king but, because of God's judgment, they had been handed over to the Babylonians.