About Number Field Counting

At this moment, I primarily study number field counting problems. These kinds of questions tend to obey a similar pattern. Fix a number field \(K\), then what can we say about the size of the set

\(\{ L/K | \) satisfies property \(P\) and \(invariant(L/K)<X\} \ \ \ \ (1)\)

as \(X\) gets large?

Two major examples come to mind:

1. Cohen-Lenstra heuristics (and other class group statistics): As \(k/\mathbb{Q}\) varies over imaginary (respectively real) quadratic extensions ordered by discriminant, what is the distribution of the \(p\)-part of the class group Cl\((k)\) for a fixed prime \(p\)? Cohen and Lenstra proposed a heuristic answer for this question in [1], giving computational and theoretical evidence that if \(p\) is odd and \(A\) is a finite abelian \(p\)-group then heuristically we expect:

\(\text{Prob}(\text{Cl}(k)_p\cong A) = \begin{cases} \frac{1}{|\text{Aut}(A)|} & \text{imaginary}\\ \frac{1}{|A|\cdot|\text{Aut}(A)|} & \text{real}\end{cases}\)

This “probability” is a limiting distribution. When we are explicit about what this means, we find the expression \((1)\) in this statement:

\(\displaystyle
\text{Prob}(\text{Cl}(k)_p\cong A) := \lim_{X\rightarrow\infty} \frac{\#\{k/\mathbb{Q}|\text{ quadratic, Cl}(k)_p\cong A, \text{ disc}(k/\mathbb{Q})<X\}}{\#\{k/\mathbb{Q}|\text{ quadratic, disc}(k/\mathbb{Q})<X\}}
\)

Studying this limit, as well as related limits for nonabelian groups or extensions of function fields, is an active area of research in this field.

2. Malle’s Conjecture: More closely aligned to the term “number field counting”, Malle studied the following set in the form of \((1)\) in [3] and [4]: Fix a number field \(K\) and a transitive subgroup \(G\subset S_n\), then define

\(
N(K,G;X) := \#\{L/K | [L:K]=n,\text{Gal}(L/K)\cong G, \mathcal{N}_{K/\mathbb{Q}}(\text{disc}(L/K))<X\}
\)

Namely, he gave evidence that

\(
N(K,G;X) \sim c(K,G) X^{1/a(G)}(\log X)^{b(K,G)}
\)

for explicit invariants \(a(G)\) and \(b(K,G)\). This is referred to as the strong form of Malle’s conjecture. The conjecture in this form is actually known to be false for the group \(G=C_3\wr C_2\) (due to Klüners in [2]). Verifying or correcting the strong form of Malle’s conjecture is a major area of research today in arithmetic statistics.

The weak form of Malle’s conjecture states that the exponent of \(X\) is correct, without stating the exact asymptotic. Namely:

\(
X^{1/a(G)} \ll N(K,G;X) \ll X^{1/a(G)+\epsilon}
\)

There are no known counterexamples to the weak form, it is proven in many more cases than the strong form, and in general there is more evidence to suggest its truth.

References


[1] H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, Lecture Notes in Mathematics Number Theory Noordwijkerhout 1983, (1984), pp. 33-62

[2] J. Klüners, A counter example to Malle’s conjecture on the asymptotics of discriminants, Comptes Rendus Mathematique, (2005), pp. 411-414

[3] G. Malle, On the distribution of Galois groups, Journal of Number Theory, 92 (2002), pp. 315-329

[4] G. Malle, On the distribution of Galois groups, II, Experimental Mathematics, 13 (2004), pp. 129-135