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

6 events

when toggle format	what		by	license	comment
Oct 14, 2017 at 15:55	vote	accept	john mangual
May 28, 2016 at 16:28	comment	added	john mangual		@DavidSpeyer I am trying to show solutions to $x^2 + y ^2 + z^2 = n$ over $\mathbb{Z}$ equidistribute over the 2-sphere as $n \to \infty$. This is too difficult. E.g. it is harder than Lagrange's 3-squares theorem. It can be solved using automorphic forms, although it relies on some challenging estimates. I am willing to take a bet there have been simplifications over the past 30 years. Here I am looking at an approach using $p$-adic analysis.
May 27, 2016 at 15:41	answer	added	paul garrett		timeline score: 6
May 27, 2016 at 15:35	comment	added	David E Speyer		If I were to guess a generalization that might work, I would try thinking of $SU(2)$ as the maximal compact subgroup of $SL_2(\mathbb{C})$, and $S^2$ as the homogenous space $\mathbb{CP}^1$. The maximal compact subgroup of $SL_2(\mathbb{Q}_p)$ is $SL_2(\mathbb{Z}_p)$, so I might think about harmonic analysis for $SL_2(\mathbb{Z}_p)$ acting on $\mathbb{P}^1(\mathbb{Q}_p)$ (which is compact). I imagine this is well understood.
May 27, 2016 at 15:31	comment	added	Julian Rosen		The $p$-adic sphere is not compact. For $p\equiv 1\mod 4$, there is a square root of $-1$ in $\mathbb{Q}_p$, and we get a sequence of points $(p^{-n},\sqrt{-1}p^{-n},1)\in S^2$ going off to infinity. For $p\equiv 3\mod 4$ it takes a little playing around with Hensel's lemma.
May 27, 2016 at 14:46	history	asked	john mangual	CC BY-SA 3.0