Having determined a source of aether for the aetheric converger, you prepare to return to Ishgard.
This quest is available if you have completed the quest Forbidden Knowledge.
Starting the Quest
Talk to Matoya in her cave in the Dravanian Hinterlands to begin the quest. There will be a cutscene. This one is important, so it is recommended to watch it either now or via the Unending Journey in an inn room.
Speak with Alphinaud
Talk to Alphinaud to proceed.
Speak with Aymeric
Go to Ishgard and use the Aetheryte or an Aethernet Shard to go to the Forgotten Knight Aethernet Shard, then go east through the door behind Firmalbert. Talk to Aymeric and there will be a cutscene.
Speak with Cid
Use an Aethernet Shard to go to the Athenaeum Astrologicum Aethernet Shard. Go south to where the orange arrow is pointing in the minimap. Talk to Cid to complete the quest. There will be a cutscene.
Next Main Scenario Quest
After you complete the quest, talk to Alphinaud to accept the next quest: Hour of Departure.