The local fields $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$, and the Adele ring $\mathbb{A}$ are all Pontrjagin dual to themselves (self duality). The consideration of the multiplicative Fourier transform of additive characters and Schwartz-Bruhat functionals $f$ with $f = \hat{f}$ (I also call this self duality) both give interesting functions.
I am thinking about which locally compact rings are Pontrjagin dual to themselves. I was hoping that there was a close relationship between self duality for locally compact rings and Adele rings.
First I want to show the connection in the "local" case: a relationship between local fields and locally compact fields with self duality. Then I would like to extend the connection to a relationship between locally compact rings with self duality and Adele rings.
Here is what I want to know:
- Can you construct all self dual locally compact fields with a self dual Schwartz-Bruhat functional using local fields and simple operations?
Let's ignore the requirement to be self-dual. What are the locally compact fields, i.e., fields with a locally compact Hausdorff topology making the field operations continuous?
Every field becomes a locally compact field when you give it the discrete topology. That is uninteresting. All other examples (locally compact fields with a non-discrete topology) turn out to be the finite extension fields of $\mathbf R$, $\mathbf Q_p$, or a Laurent series field $\mathbf F_p((t))$ for some prime $p$. And all of these are self-dual in the sense of Pontryagin duality.