Timeline for The $2\pi$ in the definition of the Fourier transform

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Mar 23, 2017 at 16:09	comment	added	WhatsUp		It is not only for esthetic reasons that one puts $2\pi$ in the exponential for $\mathbb{R}$ too. The point is that, globally one needs to deal with characters on $\mathbb{A}$, the ring of adeles, and for applications in number theory and automorphic representations, one would like to choose an additive character that factors through $\mathbb{A}/\mathbb{Q}$. If one fixes normalizations on $\mathbb{Q}_p$ as you did, then the choice on $\mathbb{R}$ is forced.
Mar 23, 2017 at 14:48	comment	added	nfdc23		The notion of "polar part" expresses in terms of coset representatives the canonical identification of $\mathbf{Q}_p/\mathbf{Z}_p$ with the $p$-primary part of the torsion subgroup $\mathbf{Q}/\mathbf{Z}$ of the group $\mathbf{R}/\mathbf{Z}$ that is identified with $S^1$ via $t \mapsto e^{2\pi i t}$. The intrinsic significance of the implicit associated self-duality of $\mathbf{Q}_p$ is to make the compact open subring $\mathbf{Z}_p$ be its own exact annihilator (and correspondingly makes the Haar measure ${\rm{d}}x$ giving it volume 1 be the associated self-dual measure on $\mathbf{Q}_p$).
Mar 23, 2017 at 14:25	history	answered	Abdelmalek Abdesselam	CC BY-SA 3.0