Improved bounds on number fields of small degree

Improved bounds on number fields of small degree, Discrete Analysis 2024:19, 24 pp.

Each number field $K$, or finite degree extension of $\mathbb Q$, has an associated rational integer, its discriminant $\mathrm{Disc}(K)$, which measures the size of the ring of integers in the field. (More precisely, $\mathrm{Disc}(K)$ is proportional to the square of the volume of the fundamental domain of the ring of integers in $K$, and its prime divisors record the finite number of rational primes that are ramified in the field.) For example, for degree two number fields, a quadratic field $K=\mathbb Q(\sqrt{D})$ with $D$ squarefree has discriminant $\mathrm{Disc}(K)=D$ if $D \equiv 1 \pmod{4}$ and $\mathrm{Disc}(K)=4D$ otherwise.

A classical theorem of Hermite states that for each fixed degree $n \geq 2$, and each fixed integer $D$, there are finitely many number fields $K$ of degree $n$ with discriminant $D$. But how many are there? For the example of quadratic fields, there is only one quadratic field with each possible discriminant. But already for degree $n=3$, the problem is quite a bit harder.

In general, the discriminant multiplicity conjecture posits that for each degree $n$ and each integer $D$, for every $\epsilon>0$ there are at most $C_\epsilon |D|^\epsilon$ number fields of degree $n$ with discriminant $D$. This problem is related to many other open problems in number theory, and currently remains out of reach for all $n>2$.

One can ask instead how many number fields there are of each discriminant, on average. For each degree $n$ and each integer $X\geq 1$, let $N_n(X)$ be the number of number fields (up to isomorphism) of degree $n$ with $|\mathrm{Disc}(X)| \leq X$. A conjecture (sometimes attributed to Linnik) predicts that for each $n \geq 2$,
$$N_n(X) \sim c_n X$$
as $X \to \infty$, for a specific constant $c_n$. For degree $n=2$, this is true, and the proof is analogous to counting the number of squarefree integers of absolute value at most $X$. For degrees $n=3,4,5$ this result is also known, and is a highly nontrivial result of Davenport and Heilbronn ($n=3$), Cohen, Diaz y Diaz and Olivier, and Bhargava ($n=4$), and Bhargava ($n=5$). For degrees $n \geq 6$, the conjecture is wide open, and is connected to many difficult questions in number theory.

One can at least ask for an upper bound for $N_n(X)$. (Lower bounds are also interesting, and difficult, but have a different flavour.) In 1995, Schmidt proved that $N_n(X)\ll_n X^{\frac{n+2}{4}}$. In 2006, Ellenberg and Venkatesh improved this to $N_n(X) \ll_n X^{\exp{c_1\exp(\sqrt{\log n}}}$ for some constant $c_1$ independent of $n$. In 2020, Couveignes again changed the qualitative shape of the record by proving that $N_n(X) \ll_n X^{c_2 (\log n)^3}$, which was soon after improved by Lemke Oliver and Thorne to $N_n(X) \ll_n X^{c_3 (\log n)^2}$ for a computable constant $c_3<1.564$. This remains the record, for large $n$.

This paper proves that $N_n(X) \ll_{n,\epsilon} X^{\frac{n+2}{4} - \frac{1}{4n-4}+\epsilon}$ for every $\epsilon>0$, a power-saving improvement over Schmidt’s bound. Independently, Bhargava, Shankar and Wang have achieved a similar power-saving improvement. Such improvements are especially noticeable in the regime of small degrees.