What is the Twisted Trace Formula? I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur.  I would like to understand the twisted trace formula, but unfortunately I never came across a good article that mainly deals with it.  For example, in Section 26 of the paper mentioned above, Arthur mentions how the formula looks like in the case of compact quotient, but that is it.  I appreciate greatly if you could answer any of the following questions.


*

*How is the kernel modified?

*How do the coarse geometric and spectral expansions look like?  How about the fine expansions?

*How can the twisted trace formula be useful?  Is there something that the twisted trace formula can detect, but the regular trace formula can not? 


Thank you!
 A: For simplicity assume that $G$ is a reductive $\mathbb{Q}$-group that is anisotropic. 
Assume that it admits an automorphism $\theta$.
Let $f \in C_c^\infty(G(\mathbb{A}))$.
One has the usual kernel 
$$
K_f(x,y):=\sum_{\gamma \in G(F)}f(x^{-1}\gamma y).
$$
The trace formula is an expression for 
$$
\mathrm{tr}R(f):=\int_{G(\mathbb{Q}) \backslash G(\mathbb{A})}K_f(x,x)dx
$$
The twisted trace formula is an expression for
$$
\mathrm{tr} R(\theta^{-1}\circ f):=\int_{G(\mathbb{Q}) \backslash G(\mathbb{A})}K_f(x,{}^{\theta}x)dx.
$$
The point is that if one expands $\mathrm{tr} R(\theta^{-1} \circ f)$ in terms of cuspidal automorphic representations, then only those representations that are isomorphic to their $\theta$ conjugates contribute, for the same reason that the trace of an $n \times n$ matrix associated to a cyclic permutation of order $n$ vanishes. Thus the twisted trace formula isolates only part of the spectrum.  One then tries to compare this part of the spectrum to spectra of other groups and thereby deduce cases of Langlands functoriality.  This is the ultimate goal of what is known as twisted endoscopy.  
The morning seminar (arXiv:1204.2888), as reproduced by Labesse and Waldspurger, is a high-level reference for this.
