Papers containing Ihara avoidance arguments

Question

I am trying to understand some of the recent research in number theory. There is apparently a certain lemma, called Ihara's lemma, which can be established in some contexts and is unknown in other contexts. Occasionally, one can still prove its consequences unconditionally. This acrobatics is called Ihara avoidance. What are some important papers containing arguments like this? Also, what are the papers containing Ihara avoidance-type argument that are technically easy to understand?

I feel like I will never be able to penetrate this sea of indices so if there is some easy paper which relies on that idea, I could try to understand it well and other papers then would become less scary.

Answer 1

For a gentle(-ish) introduction to the "Ihara avoidance" method, you might want to consult the notes of Toby Gee's course on modularity lifting from the 2013 Arizona Winter School, www2.imperial.ac.uk/~tsg/Index_files/ArizonaWinterSchool2013.pdf, where the method is worked out carefully for n = 2 in order to prove a modularity lifting theorem for Hilbert modular forms. Section 3.35 explains the relevant local deformation conditions, and section 5.5 uses them as tools in a global patching argument to prove modularity of some 2-dimensional representations.