Timeline for harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?

5 events

when toggle format	what		by	license	comment
Oct 14, 2017 at 15:55	vote	accept	john mangual
May 28, 2016 at 16:20	comment	added	john mangual		$SL_2$ or $GL_2$ should keep me busy for a long time. I also found these notes of Bump, suggesting this topic is pretty well-understood. Keep in mind though my goal is not to be terribly general or modern. My goal is to solve a problem using the limited, fairly old-fashioned methods available too me.
May 28, 2016 at 15:06	comment	added	paul garrett		$SO(\mathbb Q_p)$ is "split" means that it is isomorphic to $\mathbb Q_p^\times$, while if it is non-split, then it is isomorphic to the norm-one elements in a quadratic extension of $\mathbb Q_p$. If the quadratic form being preserved by the group is $x^2+y^2$, then when there's a $\sqrt{-1}$ the group is split, otherwise not, because of the way $x^2+y^2$ factors. For $SO(3)$ with quadratic form $x^2+y^2+z^2$, there is always a non-trivial $0$ unless $p=2$, so inside $SO(3,\mathbb Q)$ there's always a split $SO(2)$. Isogeny is a homomorphism with finite kernel and cokernel.
May 27, 2016 at 22:30	comment	added	john mangual		My, what a nice collection of notes you have. What does "split" mean here? Why are $O(2, \mathbb{Q}_p)$ and $SO(3,\mathbb{Q}_p)$ split. And what is "isogeny"?
May 27, 2016 at 15:41	history	answered	paul garrett	CC BY-SA 3.0