In Alduin's Bane, you read the Elder Scroll near the Time-Wound, and learned the Dragonrend Shout in a vision of the Nord heroes fighting Alduin in the distant past. When you returned from the vision, Alduin attacked you, and you used Dragonrend to bring him down and defeat him.
Open your quest journal:
After I defeated Alduin on the top of the Throat of the World, he fled from me. I need to locate his lair so I can finish him off, once and for all.
Talk to Paarthurnax OR talk to Arngeir OR Talk to Esbern.
Quest will unwind a bit differently depending on you you talk to. This walkthrough includes instructions for talking to ALL three NPCs, but remember you just need to speak to ONE of them.
The fastest way to complete this objective is to talk to Paarthurnax, since after just finishing "Alduin's Bane", he is standing right near you. Approach Paarthurnax and speak to him. He will tell you that you truly have the voice of a Dovah, and that Alduin's allies will think twice before attacking you after this victory. Tell him it isn't really a victory, since Alduin escaped. Tell him you need to find out where Alduin went. He will tell you that one of his allies, Motmahus, could tell us. But it will not be easy to force a dragon to betray Alduin, But the palace in Whiterun, Dragonsreach, could help you. It was originally built to house a captive Dovah. A fine place to trap one of Alduin's allies. Tell him that the Jarl of whiterun might not think so. He will tell you that his is sure that you can convince the Jarl of the need.
If you choose to talk to Arngeir, fast travel to High Hrothgar. Locate Arngeir either inside the temple or meditating in the courtyard. Talk to him. He will say he heard the Dragonrend shout from there and asks you if you defeated Alduin. Tell him that yes, but he escaped, you need to find out where he went. He will tell you that the old tales say that Alduin is able to travel to Sovngarde to devour the souls of the dead. But it doesn't say how he does it. Tell him one of his dragon allies could tell us. But how do you force a dragon to talk to you? There is one possibility, but the Jarl will never agree. Ask him what he is talking about. He will tell you that Dragonsreach, the Jarl of Whiterun's palace, was built to house a captive dragon. Tell him that you thought there were no dragons until Alduin came back. He will tell you that Dragonsreach was built a thousand years ago, in the days of Olaf One-Eyed were dragons could still be found at the peaks of mountains, before the blades wipe them out. If you can lure a dragon into Dragonsreach, with the Jarl's cooperation, you might be able to trap the dragon and force him to tell you how Alduin travels to Sovngarde
Fast travel to the Sky Haven Temple. Enter the temple, then take the door behind Alduin's wall and exit to Skyrim. Locate Esbern under one of the stone structures. He will ask you if you found the Elder Scroll and what happened when you took it to the Throat of the World. Tell him you defeated Alduin, but he escaped. You need to find him. Esbern will tell you that Alduin must have returned to Sovngarde to feed on the souls of the dead. And if you don't find him soon, he will return stronger then ever. Tell him that you know, and that's why you need to find him. Esbern will say Alduin must have some means of traveling to Sovngarde and back but he has no idea what it is.
Tell him that his dragon allies must know where this portal is. He will ask you if you know Jarl's palace in Whiterun. Ask him if he means Dragonsreach. It was originally built to hold a captive dragon. So if you could lure a dragon into Dragonsreach, you could trap him there. But it might be difficult to persuade the Jarl to use his palace as a dragon trap.
But there is another problem, a serious one. Esbern has discovered who the leader of the Greybeards really is. Tell him it turns out he's a dragon. But he helped you. But Esbern will tell you that Paarthurnax was Alduin's Chief Lieutenant in ancient times, responsible for many atrocities. The Blades have benn hunting him for centuries, but he was protected by the Greybeards, and then the Emperors. Justice demands Paarthurnax must be killed, and until then, Esbern cannot offer you his help any longer because of his oath to the Blades. Delphine also wants you to go kill Paarthurnax. Any attempts to convince them that he has changed his ways now and does not deserve to die fail in vain.
This unlocks the quest "Paarthurnax" which is an OPTIONAL quest. Doing this quest depends on whether you want to side with the Greybeards or with the Blades. If you want to side with the Greybeards then keep Paarthurnax alive. The Blades will no longer offer you assistance or additional quests. If you want to side with the Blades, then do the quest and kill Paarthurnax. The Greybeards will be upset and will no longer offer you help or give you any quests. Choose your side.
If you want to kill Paarthurnax, consult with our Paarthurnax Quest Walkthrough
In the main quest, The Fallen, the quest log updates:
"One of Alduin's dragon allies may be able to tell me where he's gone. I've learned that Dragonsreach, the Jarl's palace in Whiterun, was once used to imprison a dragon. I need to persuade the Jarl of Whiterun to help me trap a dragon in Dragonsreach.
Talk to the Jarl of Whiterun
Fast travel to Dragonsreach. Approach the Jarl. The turn of events is affected by events of the civil war. If you have already started the civil war, sided with the Imperials and wiped out the Stormcloaks, the Jarl will agree to your plan. If you joined the Stormcloaks and defeated the Imperials, the new Jarl that was appointed will also allow you to try and carry out your plan.
If the civil was has not been started (meaning you have not completed any of the Stormcloak or Imperial quests), or is yet indecisive, the Jarl (Balgruuf the Greater or anyone else that has been appointed in his place). will be reluctant to help. Tell him you need his help. You need to trap a dragon in his palace. He says he might have misheard you. Try a PERSUADE, tell him you wouldn't ask if it wasn't important. He will say that he cannot do it. Ask him about Dragonsreach, if it was really built to hold a dragon. He will tell you that yes according to legend, but he has never put it to the test. Jarl Olaf One-Eye it was, who later became the high king. He shouted it into submission in single combat atop Mount Anthor and brought it back to Whiterun. The dragon's name was Numinex, and that is his skull decorating the main hall above the Jarl's throne.
Tell him again you need his help in trapping a dragon in his palace. He will that what you are asking for is impossible and insane, to lead a dragon to the heart of the city with the threat of war on his doorstep. Tell him it is the only way to stop the dragon attacks. He will say there must be another way, the risk is too great. Tell him the threat is worse then he knows, Alduin has returned. Tell him you are Dragonborn and it is your destiny to stop Alduin. Explain to him that trapping a dragon in his palace is the only way to find Alduin before it's too late.
The Jarl will tell you he wants to help you, but he needs your help first. Ulfric and General Tulius are both waiting for him to make the wrong move, and if a dragon is attacking Whiterun, then for sure they won't sit idle and attack the weakened city. Ask him what if he wouldn't have to worry about an enemy attack. He says that in that case he would be glad to help you. But it will be hard to get both sides to agree to a truce. But if the Greybeards would be willing to host a peace council, maybe Tullius and Ulfric will listen.
Tell the Jarl to leave that to you. You will talk to Arngeir about hosting a peace council.
"If I can arrange a truce in the civil war, the Jarl of Whiterun has agreed to help me trap a dragon in Dragonsreach, then I can interrogate the dragon and learn the location of Alduin's lair so I can finish him off, once and for all."
Your objective now is to get Greybeard's help in negotitating a truce. This starts the quest, "Season Unending"