In an other fashion, you can be interested in how Fourier analysis (series decompositions, Poisson formula) is fundamental in :
- Trace formulas (kind of generalization of Poisson formula in the non-real-and-commutative case)
- Computing functional equations for zêta-functions and reaching Tamagawa numbers (those are volumes of fundamental quotient spaces in adelic settings)
- Modular and automorphic forms
For trace formulas and automorphic forms, I would say that an efficient and pleasant first lecture is H. Iwaniec, Spectral Methods of Automorphic Forms, AMS. In order to see how Fourier analysis works well in those settings, you can read Tate's thesis, it is the GL(1) case, available in Cassels-Frohlich or in Lang, Algebraic Number Theory, Springer GTM.
For Tamagawa numbers, the book of Vignéras, Arithmétique des algèbres de quaternions, Springer LNM, is a very nice reference. It is more or less translated in Reid-MacLachlan, The arithmetic of Hyperbolic 3-Manifolds, Springer GTM.
Hoping you could uncover those lovely topics ;)