New main objective: Find the Gauldur Amulet Fragment in Geirmund's Hall
Geirmund's Hall can be found east of Ivarstead, on a small island in the middle of the lake. Head inside the catacombs, kill two Skeevers and jump down the large shaft in the middle of the cave. Get out of the water and open the door behind which two Frostbite Spiders await you. Kill them and strip the nearby rooms of all treasures (look out for traps!). Inside the next room you will come across two Draugrs and a puzzle pattern: a hawk and fish on the left and on the right a fish and snake.
Head onwards, collecting treasure on your way (when you take the Soul Gem, a trap will fall down on you) and push the pillars so that they show a hawk, fish, snake and fish (it's a reflection of the pattern you have seen before - it's copied according to the corridor sides). Afterwards pull the lever by the door and enter the next room.
Defeat a couple Draugrs inside the room, approach the platform and take the Lord Geirmund's Key from the dead man's hand (screen above). Move onwards, kill some more enemies and approach the drawbridge. On the right side of the door you will find a lever - don't pull it, or you will be stabbed by spikes. The mechanism which will lower the bridge can be found behind your back, high on the wall. Pull it and go across the bridge.
Inside the corridor in front of you there are some Draugrs and swinging blades which can easily slash you in half. On the other side of the tunnel there's an undead archer, so it's best to use the Whirlwind Sprint Shout and quickly attack him (screen above). In the next room you will encounter you main enemy - Sigdis.
Get onto the platform with the coffin in which the son of Gauldur is sleeping. He will wake up and teleport behind you. On the three pedestals in the middle of the pond, not one but three copies of Sigdis will appear! Two of them are just magical copies, though they can still damage you. You can kill them with one hit, through the champion itself won't fall so easily. Therefore the easiest way of defeating him would be staying at safe distance and shooting him with a bow.
Note that Sigdis' copies don't have horns it their helms - using this you can identify them. If you happen to kill one of them, the champion will revive him and change his position (which will make the whole thing more difficult). If you keep shooting at the enemy long enough, you should be eventually able to defeat him. Take the Gauldur Amulet Fragment and Gauldur Blackbow (a very powerful artifact) from his body. Afterwards head out of the room and take everything you find useful from the nearby chest. Having done that, you can return to the surface.
New main objective: Find the Gauldur Amulet Fragment in Saarthal
Notice! In order to be able to enter Saarthal, you first need to complete the College of Winterhold quest First Lessons. The whole process of finding Gauldur Amulet Fragment is strictly connected with the College quest Under Saarthal (descriptions of both can be found in this guide). Head to the proper chapters of this guide and after obtaining the amulet fragment, return to Reforge the Gauldur Amulet found below.
New main objective: Reforge the Gauldur Amulet
Having obtained all three amulet fragments, you now have to connect them. It cannot be done at any blacksmith, but only in the cave under Reachwater Rock. Reaching it can be quite problematic - you need to head out of Markarth and go along the river until you reach a bridge below a waterfall. Afterwards go off the path and climb to around the middle of the ledge, from the right side. Below the waterfall you will find a small cave entrance (screens below).
As you enter the cave, note the corpse by the tree and the Emerald Dragon Claw lying on the pedestal. Approach the Nordic gate and set the rings as seen on the claw (i.e. bear, fish, snake). The door will open and you will be able to move on.
Inside the next room you will come across yet another door. This time in order to activate them you have to use the Ivory Dragon Claw that you found in Folgunthur in the beginning part of this quest. In accordance with the symbols on the artifact, the proper combination is: hawk, hawk, dragon.
As you enter the next room, you will see three pedestal on which you have to place the amulet fragments. Do it and ghosts of all three sons of Gauldur who you had killed will appear. It seems they won't give the artifact without a fight and you will fight them one last time.
You need to fight all brothers once again - luckily they don't attack all at once, but one after another. The first one will attack you with his powerful sword and lots of weak minions. Try to keep him at distance and shoot at him as often as possible - with the champion dead, his servants will die as well. The next brother will keep teleporting throughout the room and create his clones. Play roulette and shoot them with a hit-or-miss tactic (charging with a sword isn't suggested, as you would probably get shot dead). The last of the enemies will cast powerful spells and appear in random areas throughout the location - you should attack him from a proper distance as well.
With all of Gauldur's sons dead, he will appear in the room and rejoin the three pieces of the amulet. This way you will gain a ready item which increases your magicka, health and stamina by 30 points. It's by far one of the most powerful items in the game. At this point the Forbidden Legend quest comes to an end.