Just wanted to draw attention to this very nice exchange on Math Overflow. Matt Emerton remarks that the different of a number field is always a square in the ideal class group, and asks: is there a *canonical* square root of the ideal class of the different?

What grabs me about this question is that the word “canonical” is a very hard one to define precisely. Joe Harris used to give a lecture called, “The only canonical divisor is the canonical divisor.” The difficulty around the word “canonical” is what gives the title its piquancy.

Usually we tell students that something is “canonical” if it is “defined without making any arbitrary choices.” But this seems to place a lot of weight on the non-mathematical word “arbitrary.”

Here’s one way to go: you can say a construction is canonical if it is invariant under automorphisms. For instance, the abelianization of a group is a canonical construction; if f: G_1 -> G_2 is an isomorphism, then f induces an isomorphism between the abelianizations.

It is in this sense that MathOverflow user “Frictionless Jellyfish” gives a nice proof that there is no canonical square root of the different; the slick cnidarian exhibits a Galois extension K/Q, with Galois group G = Z/4Z, such that the ideal class of the different of K has a square root (as it must) but none of its square roots are fixed by the action of G (as they would have to be, in order to be called “canonical.”) The different itself *is *canonical and as such is fixed by G.

But this doesn’t seem to me to capture the whole sense of the word. After all, in many contexts there are no automorphisms! (E.G. in the Joe Harris lecture, “canonical” means something a bit different.)

Here’s a sample question that bothers me. Ever since Gauss we’ve known that there’s a bijection between the set of proper isomorphism classes of primitive positive definite binary quadratic forms of discriminant d and the ideal class group of a quadratic imaginary field.

Do you think this bijection is “canonical” or not? Why?

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I think the use of the word canonical in mathematics is a blend of the usage in standard English together with “preserved by isomorphisms”. To take a lower level example than those discussed in your post, I would be happy to talk of the “canonical” isomorphism Z–>Z (Z=integers) of abelian groups. Implicitly, here, I am thinking of Z as the free cyclic group with privileged generator 1; as such it has no isomorphisms, and the map sending 1->1 is “canonical”. On the other hand, the map sending 1 to -1 is also preserved by automorphisms. Of course, what is happening in this case is that we secretly recognize Z as a ring, and so have a psychological preference for the first map. So we call it canonical not only because it is preserved by automorphisms, but also because it is, colloquially, the “map we obviously would think of”. Trying to pin down a “definition” of canonical is the fundamental error of analytic philosophy — the conceit that words have a Platonic meaning — which obscures the more complicated reality of semantics.

With that in mind, I take your last question to be purely psychological, in which case, when I think of the map from the class group to binary forms, I imagine taking I = (a,b) (generators of the Z-module I) and sending I to the quadratic form N(ax+by)/N(I). I would say it is canonical in the sense that I would guess if I pick up a random book I would guess that this is the map they would use. But I could well imagine there is another “natural” definition — or perhaps a “canonical” map in the other direction whose composite is -1. To think that is “wrong” would be to fall into the trap of imagining canonical to be a purely mathematical concept. Of course, when writing a paper, one should make sure when talking about the “canonical” map from X->Y that the reader has the same map in mind as you do (e.g. when the map is canonically canonical).

I’ve been thinking of asking this on mathoverflow. This is probably because of Matt’s question, but maybe also because I recently got a referee report that told me half my canonicals should be naturals and half my naturals should be canonicals.

1. Canonical=functorial: There is a canonical map from a vector space to its double dual because it prolongs to a map of functors. I’d probably say this is what natural means too. A weaker version than functorial would be invariant under isomorphisms.

2. Canonical=definable: Something is canonical if it’s possible to write down a definition for it. So the group Z has two canonical generators, but a general infinite cyclic group has none. And R has a canonical algebraic closure (many, in fact), but a general field presumably doesn’t. Not too interestingly, Z/pZ also has a canonical algebraic closure. (Fix your favorite ordering on the polynomials and keep adjoining formal roots, in the usual way, of the first polynomial that doesn’t yet have a root.) I’m not really one for holding onto quotes from my old teachers, but Givental had a good one: “What means canonical? It means I tell you how to do it.”

3. Canonical=standard: This is probably closer to some original meaning. So then R^n has a canonical basis, the one everyone always takes, and i is the canonical complex square root of -1. Also canonical forms.

4. Canonical=something someone already dubbed canonical: This is pretty close to the previous one. For instance, probably some book somewhere states explicitly that the canonical map G–>G/N to a quotient group is to be called the canonical map.

5. Canonical=unnamed part of some structure: A vector bundle over X is defined to be a map E–>X such that… Then if you’re talking about a vector bundle E over X, you might later refer to the canonical map E–>X. Similarly, in a tensor category you could talk about the canonical map (A*B)*C –> A*(B*C). Such a map is part of the definition of a tensor category.

6. Canonical=induced by the evident universal property. Given sets X and Y, you have the canonical inclusion from X to the disjoint union.

That’s all I can think of for now. I think my current usage is moving more towards 4 and 5, probably because the other three have precise names I wrote above. Note that some of these work better with the definite article, and some with the indefinite.

I think that the bijection between ideal classes and quadratic forms is canonical. An ideal class in a quadratic number field is a lattice in Z^2. When the quadratic field is imaginary, this lattice gives rise to a torus with a unique conformal structure. Conformal structures of tori are defined by points in moduli space, H^2/PSL(2,Z). On the other hand, a quadratic form with negative discriminant corresponds to a lattice point (a,b,c) in Z^3 such that the discriminant b^2-4ac is negative. The automorphisms of this quadratic form are PSL(2,Z) again, with its 3-dimensional representation (corresponding to linear change of bases of the quadratic form). Projectivizing, the point gives rise to a unique point in the hyperbolic plane, up to the action of PSL(2,Z) (although this is the Klein model, instead of the Poincare model, of hyperbolic space). So the correspondence is natural in this geometric sense, and arising from a special isomorphism between Lie groups (sl(2,R) and so(2,1)).

Melanie Matchett Wood submitted an article on Gauss composition over schemes recently that gives an interpretation of both sides as quotients of affine three-space by equivalent actions of GL(2) x GL(1). Her construction employs a choice of equivalence, but I couldn’t tell if the discriminant adds enough rigidity to the moduli problem as stated to make the correspondence canonical (in the “unique up to unique isomorphism” sense).

You’re way ahead of me, Scott, I was going to blog about Wood’s paper for this very reason. Will get to it in the next few days. Unless you want to!