Elliptic curves over Q(i) are potentially automorphic

This is potentially big news! Announced by Richard Taylor (yes, that Richard Taylor) at the Joint Mathematics Meeting earlier this month.

https://totallydisconnected.wordpress.com/2017/01/09/elliptic-curves-over-qi-are-potentially-automorphic/

I'm interested in knowing what the property of being automorphic means, in terms similar to what it means to be modular, if that's possible.
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