# How to compute the asymptotic constant for the count of $S_3$-sextic number fields?

I am currently reading this paper counting $$S_3$$-sextic fields

• Manjul Bhargava and Melanie Matchett Wood, The density of discriminants of $$S_3$$-sextic number fields, Proc. Amer. Math. Soc. 136 (2008), 1581-1587. doi:10.1090/S0002-9939-07-09171-X

I'm trying to verify how they obtained the asymptotic constant for $$p = 2, 3$$.

My attempt:

$$c_2 =\sum_A \mu_2(A) \frac{D(A)}{D(\bar{A})^{1/3}}$$ where the sum $$A$$ is over all etale cubic $$\mathbb{Q}_2$$ algebra $$A$$, $$\mu_2(A)$$ is the relative density as defined on the paper on page 3 (or 1583), and $$\bar{A}$$ is the $$S_3$$ closure of $$A$$. To compute $$\mu_2(A)$$, I referred to Davenport-Heilbronn paper

in particular Lemma 18 and Lemma 19 in the arXiv version. To look up all the possible extensions, I went over to LMFDB at https://www.lmfdb.org/padicField/.

For example, if $$A$$ is the cubic extension of $$\mathbb{Q}_2$$ with ramification index 3 and inertia degree 1, the relative density would be $$\frac{\mu(\mathcal{U}_p(1^3))}{\mu(\mathcal{U}_p)}$$. The discriminant of $$A$$, $$D(A)$$, would be $$2^2$$ since the discriminant exponent $$c=2$$, and the discriminant of the $$S_3$$-closure of $$A$$ would be $$2^4$$. I computed this by looking at the Galois splitting model provided in LMFDB, and finding the discriminant of the $$p$$-part of splitting field.

Doing this for all cubic etale algebra of $$\mathbb{Q}_2$$, I have that $$c_2 = \frac{\mu(\mathcal{U}_p(111))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(12))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(3))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(1^21))}{\mu(\mathcal{U}_p)}\left(2\frac{p^2}{p^{2/3}} + 4\frac{p^3}{p^{3/3}}\right) + \frac{\mu(\mathcal{U}_p(1^3))}{\mu(\mathcal{U}_p)} \frac{p^2}{p^{4/3}} =\frac{1+2p^{1/3}+4p+p^{-4/3}}{1+p^{-1}+p^{-2}}$$

The 2 and 4 are because there are 2 quadratic extensions of $$\mathbb{Q}_2$$ with $$e=2$$ and $$c=2$$, and 4 quadratic extensions of $$\mathbb{Q}_2$$ with $$e=2$$ and $$c=3$$. I thought maybe I should replace 2 with 2/6, and 4 with 4/6 since there are 6 quadratic extensions with $$e=2, f=1$$, but it still does not work. Doing the same for $$p=3$$, I have

$$c_3 = \frac{\mu(\mathcal{U}_p(111))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(12))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(3))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(1^21))}{\mu(\mathcal{U}_p)}\left(2\frac{p}{p^{1/3}}\right) + \frac{\mu(\mathcal{U}_p(1^3))}{\mu(\mathcal{U}_p)}\left(2\frac{p^3}{p^{7/3}} + 3\frac{p^4}{p^{4/3}} + \frac{p^4}{p^{8/3}} + 3\frac{p^5}{p^{11/3}}\right)$$

1. Is my computation of the relative density correct? Or have I misunderstood the definition from the paper?

2. To compute the discriminant of the $$S_3$$ closure of a cubic extension $$A$$ of $$\mathbb{Q}_p$$, is it right to compute the $$p$$-part of the discriminant of the splitting field of the Galois splitting model of the particular cubic extension $$A$$?

These are vastly different from what they had on their paper. Obviously, I have a severe misunderstanding/error in my calculation and I would appreciate if someone can point out where my mistakes are in particular.

• Please use a high-level tag like "nt.number-theory". I added this tag now. Feb 11 at 20:46
• @GHfromMO Sorry, first time. And thanks! Feb 11 at 20:56
• Why was this downvoted? At first glance this seems like a pretty appropriate question. Feb 18 at 23:21
• I"m not an expert on this, I just tidied up the citations to be more informative and stable. I can't judge the number theory content so I'm not sure what precisely is wrong with it. Feb 19 at 0:38
• @DavidRoberts Thanks! Sorry you had to do that. I'll be very clear and explicit next time. Feb 19 at 3:12

I will do a worked example for the tame case and show that it agrees with the formula from their paper. You should then be able to adapt this to the case $$p=2,3$$.

Let $$p > 3$$. I use formula (7) from the paper. We need to show that $$\sum_{[L:\mathbb{Q}_p] = 3} \frac{1}{|\mathrm{Aut}(L)|D(\bar{L})^{1/3}} = 1 + 1/p + 1/p^{4/3}$$ where $$\bar{L}$$ denotes the $$S_3$$-closure of $$L$$ and $$D$$ its discriminant. Here $$L$$ runs over cubic etale $$\mathbb{Q}_p$$-algebras. The local factor $$c_p$$ in Theorem 2 is obtained by multiplying this by $$(1-1/p)$$.

We consider each case in turn.

1. $$L= \mathbb{Q}_p^3. |\mathrm{Aut}(L)| = 6, D(\bar{L}) = 1.$$
2. $$L = \mathbb{Q}_p \times K$$, $$K$$ quadratic. $$|\mathrm{Aut}(L)| = 2, D(\bar{L}) = 1$$ if $$K$$ is unramified and $$D(\bar{L}) = p^3$$ if $$K$$ is ramified.
3. $$L$$ cyclic cubic. $$|\mathrm{Aut}(L)| = 3, D(\bar{L}) = 1$$ if $$L$$ is unramified and $$D(\bar{L}) = p^4$$ if $$L$$ is ramified.
4. $$L$$ non-Galois cubic. $$|\mathrm{Aut}(L)| = 1, D(\bar{L}) = 1$$ if $$L$$ is unramified and $$D(\bar{L}) = p^4$$ if $$L$$ is ramified.

We work this all out for the case $$p = 5$$. There is a unique unramified quadratic extension and two ramified quadratic extensions. There is a unique cyclic cubic extension, which is unramified, and a unique non-Galois cubic extension, which is ramified. (You can get all this from the LMFDB.)

All together we obtain: $$\frac{1}{6} + \left(\frac{1}{2} + \frac{2}{2p}\right) + \frac{1}{3} + \frac{1}{p^{4/3}} = 1 + \frac{1}{p} + \frac{1}{p^{4/3}}$$ as required.

(The $$S_3$$-closure is a bit different from the Galois closure, see the paper for more details. This explains some of the strange exponents in $$p$$ which appear above, as the index of the subgroup in $$S_3$$ plays a role in the exponent. This may be where your mistake lies. These exponents should be checked again carefully!)

• Hi, thank you very much for pointing out where I made the mistake. It did not occur to me to use the formula from (7). I went over the construction for the $S_K$-closure of an etale algebra but frankly, the structure of is a bit over my head. To compute discriminant of a number field, I know we generally find the $\text{Tr}(x_ix_j)$ for a set of integral basis, but I'm confused how one does that when the basis is a set of tensors. Could you please refer me to a source where I can learn how to compute discriminants such as the above? Thank you in advance! Feb 21 at 16:25
• I'm not entirely sure I'm afraid. I came up with the discriminant exponents by looking at Section 2.1 of Bhargava's "Higher composition laws III: The parametrization of quartic rings". This explains to the definition of the $S_3$-closure and some examples. You shouldn't try to work directly with the definition of the discriminant, but rather relate the discriminant of the $S_3$-closure to the cubic algebra, in a similar way to Bhargava and Wood do at the start of Section 2. Feb 22 at 11:21