p-adic numbers of norm 1 Hello everyone,
The complex numbers of modulus 1 may be seen as the real points of the scheme 
$\mathrm{Spec}\ \mathbb{Q}[X,Y] / (X^2+Y^2-1)$.
Can elements of norm $1$ in $\mathbb{C}_p$ be seen as the set of $\mathbb{Q}_p$-points of an affine scheme?
A related question is: what are the $\mathbb{Q}_p$-points of $\mathrm{Spec}\ \mathbb{Q}[X,Y] / (X^2+Y^2-1)$ ?
Best,
Sibhwa
 A: The difference is that $\mathbb{C}/\mathbb{R}$ is a finite extension.  This means that you can write a general element of $\mathbb{C}$ as $x+iy$ with $x,y \in \mathbb{R}$, take its norm to get $x^2+y^2$, and set that equal to 1; the resulting equation defines an affine variety $T$ over $\mathbb{R}$, whose $\mathbb{R}$-points correspond to the complex numbers of norm 1.
The variety $T$ is more than just a variety: the multiplication on $\mathbb{C}$ turns it into an algebraic group, and it is called the norm torus for the extension $\mathbb{C}/\mathbb{R}$.  An analogous construction works for any finite extension.  However, you ask about $\mathbb{C}_p / \mathbb{Q}_p$, which is an infinite extension (and not even algebraic); it's not clear how you should define the norm for an infinite extension.  Of course, $\mathbb{C}_p$ is a vector space over $\mathbb{Q}_p$, though a hugely infinite-dimensional one, so you can think of elements of $\mathbb{C}_p$ as being represented by the $\mathbb{Q}_p$-points of an enormous affine space if you like, though it's not clear whether this is ever going to be helpful.
As for your second question: for any field $K$ containing $\mathbb{Q}$, the $K$-points of your variety represent elements of $K[t]/(t^2-1)$ of norm 1.  If $-1$ is a square in $K$, then this algebra is isomorphic to $K^2$; otherwise, it is the field $K(\sqrt{-1})$.
