To start with, players looking to start this quest should know that to initiate it, they must first have completed "A Relic Reborn" for the specified weapon they are attempting to upgrade and have it upgraded to Zenith status. For those that do not yet know where "A Relic Reborn" is started, venture to the North Shroud (Fallgourd Float teleport) and proceed to X30, Y20 and speak with Gerolt. He will set you along the path to getting that completed.
Once the player has completed "A Relic Reborn" and gotten their relic weapon to Zenith status, proceed back to Gerolt and he will point you to another NPC to get "Up in Arms" started. I won't post where the NPC is, because he is quite literally a few meters behind you after you talk to Gerolt again .
From there, you will be given a list of where each Atma stone can be farmed. Please take note that the list that follows is the correct list and the one in game has several typos. First of which is that it tells the player that 2 stones can be gotten from Central Thanalan when, in fact, only one exists. The stone that comes from Central Thanalan is the Atma of the Scales. The Atma of the Maiden is obtains from Central Shroud. A second typo is that one is said to exist in "Central" La Nocea. Please note that this was intended to be "Middle" La Nocea as no "Central" La Nocea exists. And lastly, the stone in the list labled "Atma of the Water Eater" is actually called (from the actual tooltip of the stone) Atma of the Water Bearer. Minor one there but thought I'd point it out anyway. The list of the stones and their locations are as follows. Also note that EVERY Fate in the specified zone has a chance to drop your stone. And as far as we know so far, the drop rate is %5 (Please correct me if I'm wrong) and there are no "special" fates designed specifically for this quest.
Southern Thanalan: Atma of the Scorpion
Eastern Thanalan: Atma of the Bull
Western Thanalan: Atma of the Twins
Central Thanalan: Atma of the Scales (There is only one stone in Central despite what the log says. Apparently the game flubbed the location. Actual location to follow.)
Central Shroud: Atma of the Maiden (this is the one that the text originally said was in Central Than as well)
East Shroud: Atma of the Goat
North Shroud: Atma of the Archer
Upper La Nocea: Atma of the Water Bearer
Middle La Nocea: Atma of the Ram
Outer La Nocea: Atma of the Lion
Lower La Nocea: Atma of the Fish
Western La Nocea: Atma of the Crab
What we know so far about the stone drops:
*You do not have to stick around for the Fate to it's entirety. Once you have participated, you can move onto the next Fate as long as there are enough people at the previous to actually complete it. Not certain at this time if you can still get a drop from a "failed" Fate. (Please confirm if you have gotten a drop from a failed one.)
*Drops have been confirmed by players for all ratings; That means Gold, silver, bronze, doesn't matter nor does it affect your % for a drop rate. Which means as long as you participate in the Fate, you will get the same chance as everyone else for your drop. This is so that, (I suspect) Fate groups don't hold a completely unfair advantage over Solo players. Also note that being in a group, or solo, your chances for a drop ARE THE EXACT SAME as everyone else.
*Changing zones after farming one zone for a specified amount of time DOES NOT affect your drop rate. (Sensing a theme here?) To cover any other further rumors, there is absolutely nothing that affects your drop rate one way or another. Let's be logical here and steer away from Pigeon Theory shall we?
If anyone has anything else to add to this section, post it here if you would kindly and I will get the necessary addition up asap. Please keep it logic based and as close to "fact" as possible. I'd like to keep away from "rumors" and such if at all possible.
After you've obtained your 12 Atma Stones (for me it took a week but there have been some players who got all of theirs in a day or two so the farm time varies.) Head back to the Elvaan NPC in North Shroud you spoke with before to get this started. Before speaking with him, make sure that you have a place holder for your weapon as you will have to unequip your relic much in the same way you did when you went to upgrade it to Zenith state. The process takes but a moment in a short cut scene and you will have your weapon back in it's Atma state. Note at this point, your base Atma will be identical in stats to your Zenith but with ilvl100 attached to it. It is not until you undergo the next step(s) (yes more grinding but not as bad imho) that you will start to see your new weapon begin to come into it's own.
The next step, after handing in your Atma Stones, will be to travel to Revenant's Toll in Mor Dhona and speak with with G'Jusana (X22, Y7) (I believe? There may be one short step before that but just follow the quest and you'll get to her in no time.) Here, she will explain roughly that there are 9 separate books for your weapon, each increasing a specific stat (For me as a Black Mage, those stats are Intelligence, Vitality, Spell Speed, and Determination). Each of these books will cost you 1500 myth tomes. After purchasing a book for the specified stat you want to increase, it will have you completing various tasks. According to said NPC, each weapon is different so the tasks therein will be different based on the Job's weapon.
However you choose to farm myth tomes, buckle in and get comfortable with it because it's going to be another arduous grind fest before you're done. If I've missed anything, or again, if anyone has anything else to add (within the realm of fact and not hearsay of course) post it here and I will attempt to update this as needed. Thanks again and I hope this has been at least somewhat helpful to you guys. As always, Happy Atma Games and may the odds always be in your favor
Edit: Added the compiled notes of what we know so far and changed set up so that everything is as up to date as possible.