Matoya locked away the research that may help you reach Azys Lla, and you will have to risk life and limb to retrieve it.
This quest is available if you have completed the quest Matoya's Cave.
Starting the Quest
Talk to Matoya in her cave in the southwest of the Dravanian Hinterlands to begin the quest.
Speak with Matoya
Talk to Matoya to proceed.
Speak with the Enchanted Broom
Go to the quest icon on the map and talk to the Enchanted Broom.
Speak with Broomsy
Go northwest and go through the tunnel to the cave's exit. Use the exit, then go east to where the orange arrow is pointing in the minimap. Go up the stone stairway, then when you reach the top, turn right, and go east to where the orange arrow is pointing in the minimap. Talk to Broomsy to proceed.
The Great Gubal Library
Use the Duty Finder or Duty Support to enter the Great Gubal Library. Please see the The Great Gubal Library Guide for a walkthrough.
After you complete the duty in the Great Gubal Library and exit the area, there will be a cutscene.
Deliver the Tome
Go west to the bridge, and when you are in the middle of the bridge, look for the stairs on the south side of the bridge. Go down the stairs, then continue west to where the orange arrow is pointing in the minimap. Enter Matoya's cave and go south to where the orange arrow is pointing in the minimap. Talk to Matoya to hand over On Aetheric Convergence and complete the quest.
Unlock Flying in The Dravanian Hinterlands
"Forbidden Knowledge" is the final quest for unlocking Aether Currents in the Dravanian Hinterlands. If you have completed the other Aether Current quests and attuned to the Aether Currents in the field in this area (which you can find using the Aether Compass in Main Menu > Duty > Collection in this area) you will be able to fly in the Dravanian Hinterlands.
Next Main Scenario Quest
After you complete the quest, talk to Matoya to accept the next quest: An Eye for Aether. 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.