Question

What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker?

(So I do not mean weaker in the sense that the Arthur trace formula implies the simple trace formula.)

Answer 1

I must have missed this question a month ago, but hopefully you're still interested.

I haven't read that particular paper, but simple trace formulas have restricted test functions, which for one can only deal with representations which are supercuspidal at some place. So any theorem you prove, say about transfer of representations, needs to be restricted to functions which are supercuspidal at some finite place. In general, there may be some other restrictions also, but that depends upon the details of the simple trace formula at hand.

Answer 2

The simple trace formula in Flicker-Kazhdan is exactly what you get when you restrict the full Arthur trace formula to "simple" test functions -- ones where many of the terms on both sides vanish. Importantly, all the complicated terms that are achieved only through regularization vanish, so that the proof of this simple trace formula is also simple.

On the spectral side, only representations with a supercuspidal component appear; on the geometric side, only regular orbits appear. Any result that needs to account for the trivial representation or the orbit {1} won't work with this simple trace formula.

Kottwitz's proof of the Tamagawa number conjecture uses a different simple version of the Arthur trace formula. This one includes the term for the orbit {1}, an essential point since that is the term with the Tamagawa number for the group.

This other simple trace formula can be stated as simply as the one in Flicker-Kazhdan, but is more general as it requires less about the test functions. Its proof requires first constructing the full Arthur trace formula and then verifying that the messy terms vanish. It still won't handle the trivial representation.