I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism `X \to Y` survives a *base change* `f:Z\to Y`, meaning that `X \times_{Y} Z \to Z` also has this property.



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> **Question:** Is the base change in number theory and derived algebraic geometry the same thing as above? What would be the examples?
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What is the base change in number theory?
In number theory, base change can also refer to an operation on automorphic representations. If L/K is an extension of number fields, and pi is an automorphic representation of a reductive group G over K, then pi should "lift" to a new automorphic representation of G over L. This is the sense of the phrase used in, e.g., Langlands' book "Base Change for GL(2)". The existence of a certain kind of base change for GL(2) was used to prove the modularity of some mod 3 Galois representations, which in turn played a role in proving Fermat's last theorem.


