Why are quadratic residues more interesting than cubic residues?

Question

There is a large theory of quadratic residues. As far as I know there is no comparably large theory of cubic residues or history of studying them. A search on this site finds hundreds of questions about quadratic residues (which even has its own tag, quadratic-residues), and only 31 about cubic residues.

What mathematical features of quadratic residues lead to their having an important and rich theory, whereas there seems to be relatively little to say about cubic residues?

I have a couple of thoughts:

• The Brahmagupta-Fibonacci identity ensures that quadratic forms have interesting multiplicative properties; there is no corresponding identity for cubes.
• The Euclidean metric involves second powers and not third powers, so questions about geometric properties of $\Bbb Z^n$ lattices are more likely to depend on properties of second powers than third powers.

But perhaps these are two faces of the same underlying issue. I think they tie together in the fact that while $|z|=\sqrt[3]{x^3+y^3}$ can be used as a norm on $\Bbb C$, it lacks the important property that $|z_1z_2| = |z_1||z_2|$.

Still I feel like I'm missing the bigger picture. What's the bigger picture?

Answer 1

Quadratic residues appear more interesting because elementary number theory is about $\mathbf Z$ and in $\mathbf Z$ the only units are $\pm 1$. We can do things with cubic and quartic residues in $\mathbf Z$, but it's awkward at times because we don't have the cubic or quartic roots of unity in $\mathbf Z$.

If you do number theory in $\mathbf Z[\omega]$ (where $\omega$ is a nontrivial cube root of unity) or $\mathbf Z[i]$ then you could say interesting things about cubic residues and quartic residues, respectively. That's where you have the cubic and quartic reciprocity laws.

To have nice general aspects of $n$-th power residues you want to work in a setting where you have the $n$th roots of unity.

The ubiquity of quadratic forms compared to higher-degree forms is another issue: see here.

Answer 2

I don't know that there's one decisive thing to say about this, so I'll toss out a bunch of observations that I think add up to some kind of picture. I think we can meaningfully broaden the question in at least two directions:

• What makes quadratic forms special compared to cubic forms, etc.? (Quadratic vs. cubic residues corresponds to the special case of understanding the quadratic form $x^2$ vs. the cubic form $x^3$ over $\mathbb{F}_p$.)
• What makes the number $2$ special, in general? See e.g. Is there a high-concept explanation for why characteristic $2$ is special? and What's so special about characteristic $2$?, and probably others.

So, here are a bunch of observations.

1. $2$ is the only prime that divides $p - 1$ for all but finitely many primes $p$; this means that for every prime $p \ge 3$ exactly half the nonzero residues $\bmod p$ are quadratic residues, and half are quadratic non-residues. So quadratic residues provide a meaningful decomposition of $\mathbb{F}_p^{\times}$ for all $p \ge 3$, whereas e.g. the theory of cubic residues is trivial for $p \equiv 2 \bmod 3$ and only interesting for $p \equiv 1 \bmod 3$.

2. The behavior of quadratic forms can be distinguished from cubic and higher forms based on the following dimension counting argument. The space of homogeneous $d$-forms in $n$ variables has dimension ${n+d-1 \choose d}$, but $GL_n$ has dimension $n^2$. The two are only comparable for $d = 2$, so it's only quadratic forms that 1) can have a reasonably well-behaved classification up to isomorphism, and that 2) have a good chance of having interesting automorphisms generically. For example, the automorphisms $O(n)$ of the Euclidean quadratic form! This dimension count heuristically suggests, and I think this is known to be true, that the automorphism group of the generic cubic or higher degree form is finite. For more on this see, for example, There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?, and also Nature of the Euclidean Norm for a discussion of what distinguishes the Euclidean norm from other norms on $\mathbb{R}^n$, in particular the $\ell^p$ norms.

3. The distinction above between quadratic and cubic+ forms manifests itself in both the classification of semisimple Lie algebras (and related classifications: compact semisimple Lie groups, reductive groups, etc.) and in the classification of finite simple groups, suggesting it is quite robust. Namely, we have infinite families of groups attached to quadratic forms (the orthogonal and unitary groups / Lie algebras, and their analogues over finite fields), but no such infinite families attached to cubic or higher forms. Some of the exceptional groups can be defined in terms of cubic forms, but none of them generalize to infinite families, suggesting the cubic forms involved are quite special (for example, $G_2$ can be defined in terms of a cubic form on $\mathbb{R}^7$ related to the octonions). In other words, both the classification of semisimple Lie algebras and the classification of finite simple groups tell us that there is no infinite family of cubic+ analogues of the orthogonal or unitary groups.

4. Related to the dimension counting argument, a nondegenerate quadratic form produces an identification of a f.d. vector space $V$ with $V^{\ast}$. This suggests more generally that quadratic things are related to dualities, and dualities are a very fundamental and common theme in mathematics. So cubic forms might be related to "trialities"; these exist but are much less common, for example the $\text{Spin}(8)$ triality (which is again related to the octonions!). If we define a triality as a suitably nondegenerate trilinear pairing between $3$ vector spaces then it turns out that over $\mathbb{R}$ they only exist in dimensions $1, 2, 4, 8$, and are closely related to the division algebras $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ in those dimensions; see the links for more.

There's probably much more to say but I think that's already plenty. I think there is quite a deep mystery here and I'd like to understand it better.