We prove a relative trace formula that establishes the cubic base change for $\GL(2)$. One also gets a classification of the image of base change. The case when the field extension is nonnormal gives an example where a trace formula is used to prove lifting which is not endoscopic.

MSC Classifications:

11F70, 11F72 show english descriptions Representation-theoretic methods; automorphic representations over local and global fields
Spectral theory; Selberg trace formula 11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11F72 - Spectral theory; Selberg trace formula

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