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 ;)

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