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



p-adic numbers of norm 1
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 $K(\sqrt{-1})$.


