# automorphic forms – The notion of smoothness in the local situation

I am reading Bump’s book on Automorphic forms and Represenations and I am able to draw a lot of paralells between the theory of $$GL(2, mathbb{R})$$ which is the infinite place and the theory of $$GL(2,F)$$ where $$F$$ is a local field which corresponds to the finite places.

There are notions of smooth functions in both the setups. In the case of $$GL(2,mathbb{R})$$, the smoothness is the usual definition (considering the differential manifold structure on $$GL(2,mathbb{R})$$). But in the local case, smoothness is defined to be locally constant. Is there any differential structure on $$GL(2,F)$$ which justifies this definition? Or in other words, what is the motivation for defining smoothness in this fashion?