Speak with Jalzahn with your Atma Zodiac Weapon equipped who will explain that in order to further enhance your weapon you need to recreate the valorous deeds of the Zodiac Braves, who were revered heroes in Thavnairian legend. These deeds are recorded in a collection of nine sacred texts called the “Trials of the Braves”. He will send you to Rowena in Revenant's Toll - who in turn will direct you to G'jusana who is nearby. After speaking to G'jusana, the quest will move to your "completed quest log".
G'jusana will sell you one book at a time for 500 Allagan Tomestone of Soldiery each. Each book contains a series of tasks to complete: 10 specific monsters to kill and 3 dungeons, 3 FATEs and 3 levequests to complete. Once all the tasks are completed in one book, return to G'jusana to buy the next one. You must finish all the tasks in one book before completing the next - however you can complete the books in any order. Each book will improve a stat on your Atma Zodiac Weapon.
Each book belong to one of four categories. Each category improves a stat on your Atma Zodiac Weapon by +2 and each category is called a different name depending on the job. However, the books themselves and the tasks inside them remain the same regardless of your job.
You can only ever own one book at a time. Each book is specific to a weapon and you must be equipped with the Atma Relic Weapon you wish to enhance when making the purchase.
Must equip the corresponding weapon in order for any completed task to be registered.
Buying a second book will automatically dispose of the first one. If the first book has not been completed it will need to be redone.
Each map has a maximum number of FATEs available at any one time. Therefore you may need to clear other FATEs in order to get the one you need to spawn.
Some FATEs will only spawn when triggered (either by talking to an NPC at the location or in a nearby location; or after the completion of a previous FATE as part of a chain).
It is not uncommon for people to ask over the zone chat when the last rare FATE has spawned.
General levequests and Grand Company levequests are given out by two different NPCs at each location. Make sure you are talking to the correct one.
Only a limited number of levequests are offered at any one time. You may need to complete some of the levequests in order to have the one you need pop up.
Your last book will still remain in your Key Item Inventory after complication. To remove it, simply return to G'jusana and select the Disposal option.
This is a good opportunity for you to train your Chocobo if you wish.