Why is the regulator of a number field normalized the way it is?

Question

The regulator of a number field is essentially the covolume of the unit group embedded into the vector space $\{(x_1, \ldots, x_{r+s}): \sum_i x_i=0\}$ under the log embedding: $$x \mapsto (\log |\sigma_1(x)|, \ldots \log |\sigma_r(x)|, \log |\sigma_{r+1}(x)|^2, \ldots , \log |\sigma_{r+s}(x)|^2).$$

But you need to take some care in terms of what the right Euclidean structure is for defining volumes in $\{(x_1, \ldots, x_n): \sum_i x_i = 0\}$. What you're supposed to do is to use the Euclidean measure induced by any coordinate projection $\{(x_1, \ldots, x_{r+s}): \sum_i x_i = 0\} \rightarrow \mathbb{R}^{r+s-1}$. Equivalently, you could use the subspace Euclidean measure, but then normalize by dividing by $\sqrt{r+s}$.

My question is why is this the best normalization?

So far the only answer I have is that for the quadratic real case this makes the regulator exactly the log of the fundamental unit, which seems a very sensible convention. I guess it also makes the analytic class number formula slightly cleaner, but it's not obvious to me why this normalization exactly comes into the class number calculation.

My apologies if this question is too elementary.

Answer 1

Let $V$ be a vector space and $W$ a subspace.

Given a volume on $V$ and a volume on $V/W$, we can define a volume on $W$. (Here "volume" = "translation-invariant measure" but no real measure theory is being used. We could also define a volume as an element of the top wedge power of the dual vector space.)

Geometrically, this says that given a basis $v_1,\dotsc, v_m$ of $W$ which extends to a basis $v_1,\dotsc, v_n$ of $V$, the volume of the parallelepiped spanned by $v_1,\dotsc, v_m$ in $W$ is equal to the volume of the parallelepiped spanned by $v_1,\dotsc, v_n$ in $V$ times the volume of the parallelepiped spanned by $v_{m+1},\dotsc,v_n$ in $V/W$. (The same thing works with any shape in $V$ whose fibers under the projection map to $V/W$ are all translates of a fixed shape in $W$.)

We can express this algebraically with top forms as well, as François Brunault explains in the comments.

For $V$ the vector space of tuples $(x_1,\dotsc, x_{r+s})$, and $W$ the subspace $\sum_i x_i=0$, it is natural to identify $V/W$ with $\mathbb R$ under the map $\sum_i x_i$ and take the standard volume on $\mathbb R$. When $x_i$ are the logs of the individual coordinates, this identification corresponds to the norm map.

The reason this is the most commonly used normalization in number theory problem has to do with the fact that the norm map is used often in number theory. In particular, the norm is used in defining the zeta function, explaining why this approach gives nice formulas for the residue of the zeta function.