Skyrim Walkthrough Part 13: Elder Knowledge

In the previous main quest, "The Throat of the World", you sought out Paarthurnax, the Greybeards' leader, on the top of the mountain. You learned that  he does not know the Dragonrend Shout needed to defeat Alduin, but he believes that there is a way to learn it from the ancient heroes who used it against Alduin.

ELDER KNOWLEDGE
I met the master of the Greybeards, the dragon Paarthurnax. He told me that the ancient Nords used an Elder Scroll to cast Alduin forward in time. If I can recover that same Elder Scroll, I may be able to use it to learn the "Dragonrend" shout directly from the ancient heroes.

Objectives
Learn the location of the Elder Scroll or (Optional) talk to Arngeir

While highly unlikely, it is possible that your character has already completed the Daedric Quest "Discerning the Transmundane", and the Elder Scroll is in your possession already. If this is the case, then you will immediately start the next quest, "Alduin's Bane"

First, return to High Hrothgar to complete the optional objective of talking to Arngeir. You could find him either inside High Hrothgar, or in the courtyard.

Next, fast travel to Winterhold. If you have done the College of Winterhold quests previously, you can instead fast travel directly to the College of Winterhold. If you have never been in Winterhold, fast travel to Whiterun and take the carriage to Winterhold.

In Winterhold, head North towards the college. You will run into Faralda who does not let you in until you can prove to her that you know magic. She will give you a spell to cast as a test. But instead of casting the spell she wants, just use your fire breath shout on the seal near her. This will convince her that you are worth of joining the College of Winterhold. This college has a separate quest line which we recommend doing, even if you are not a mage. Please refer to our College of Winterhold: First Lessons Walkthrough if you are interested in doing these side quests.

Climb up to the college and enter the Hall of the Elements. Then take the door to the right to enter the Arcanaeum. Talk to Urag Gro-Shub. Tell him you are looking for an Elder Scroll. Ask him to tell you more about the Elder Scrolls. He will tell you that an Elder Scroll is an instrument of immense knowledge and power. It's a reflection of all possible futures and all possible pasts.

Ask him if he has an Elder Scroll you could use. He said that even if he did, he wouldn't let anyone lay a finger on it. Ask him if that includes Dragonborn. He will realize that you are the one the Greybeards were calling, and will agree to help you.

He will retrieve some books from the bookcases. On his desk, read the books "Effects of the Elder Scrolls" and  "Ruminations on the Elder Scrolls". This starts the Daedric Quest "Discerning the Transmudane". During that quest you will find the Elder Scroll. At this point, follow our Discerning the Transmundane Walkthrough to obtain the Elder Scroll. Once you are done with that, fast travel back to Paarthurnax on the top of the mountain at the Throat of the World and start the quest "Alduin's Bane".