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There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions surrounding the Langlands program, but one of the major tools there, the trace formula, is much better developed in the characteristic $0$ zero situation than in characteristic $p$.

Probably the main reason for this is that most people in this field are more interested in the characteristic $0$ setting. However, I have also heard that there are serious technical difficulties in implementing the trace formula over function fields, which do not appear over number fields. Is this the case, and if so, what are they?

For concreteness, let me mention two specific well-known problem where the function field case appeared more difficult.

  1. Weil's Tamagawa Number conjecture was proved over number fields many years ago, using the trace formula, while the analogous statement over function fields was not proved until quite recently by Gaitsgory-Lurie, using rather high-powered machinery.

  2. The base change fundamental lemma was proved in characteristic 0 by (cumulative) work of Kottwitz, Clozel, and Labesse, using the twisted trace formula. As far as I am aware, there is still no proof in characteristic p (though perhaps model theory can be used to deduce the result for sufficiently large $p$ relative to the group).

For either of these examples is there, in principle, any reason one could not adapt the characteristic 0 proof to characteristic p?

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