1. Her presence is what causes the fallout between the Avengers and X-Men; Cyclops and his teammates want to safeguard Hope, while Captain America and the Avengers want to use her to safeguard the planet from the Phoenix. That's the initial thrust of Avengers vs. X-Men.
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2. Early into his run of Avengers Vol. 4, Brian Michael Bendis whisked several characters into a dystopian future. An elderly Iron Man revealed to his younger counterpart an extensive timeline of world disasters. The timeline included many events that have already unfolded (Siege, Chaos War), some that had yet to unfold at the time (Fear Itself, The Galactus Seed), and many that may or may not occur in the future.
The timeline culminates with an event called "The Ultron War" and the resulting conflict between Kang and Ultron that created this dystopian future. It's difficult to tell right now if this timeline offers any clues to Avengers vs. X-Men, but it is clear that even that big event is only one part of the equation in 2012. Bendis further teased The Ultron War in Avengers #12.1 this year, with the promise that the story would begin in force in 2012. That fact was reiterated when Marvel announced that their Free Comic Book Day release for 2012 will be a reprint of Avengers #12.1.
The Ultron War will be another of Marvel's big storylines in 2012. Will it play into Avengers vs. X-Men? We've seen Marvel's writers tease an upcoming conflict in several books that could involve either one of these stories. Most notably, the image of robotic tripods has cropped up in several books. They appeared in the first arc of Avengers Vol. 4 as the traditional Martian tripods from War of the Worlds and the Killraven comics. They appeared in Tony Stark's dream world in the "Stark Disassembled" arc of Invincible Iron Man. They appeared in Steve Rogers' vision of the future in Captain America: Reborn. Why does this threat keep reappearing in separate books and from separate writers (all of whom are contributing to Avengers vs. X-Men)?
These two events may or may not have any direct ties. But one thing we find interesting about both is that they take place in 2012. 2012 is the final year on the Mayan Long Count calendar, a fact that has famously invited all sorts of doomsday prophecies and conspiracy theories.
However, the Mayan stance about 2012 is not so much about death and destruction but change. Death and rebirth follow a perpetual cycle, an idea that has also been explored in recent Thor storylines. This applies nicely to the idea of the Phoenix Force. The end of the world is already a possibility, so could Avengers vs. X-Men or another event deal with the 2012 prophecies?
So could the Avengers want Hope to take control of the Phoenix Force to possibly stop Ultron? Maybe the way of avoiding this Ultron War that's been hinted at for ages now is a DC-esque reboot in 2012. Marvel have been building this up for a while..