2-torsion in class groups of cubic fields

Question

I was wondering if there are good bounds for the $p$-parts of the class group of a number field $F$ in terms of its discriminant $D_F$. More precisely, the bound for the order of the full class group of $F$ is of order $\sqrt {D_F}$ and I was wondering whether for a fixed prime $p$ there is a bound for the $p$-torsion of order $D_F^\delta$ for small $\delta > 0$ (ideally arbitrarily small). I am especially interested in $p=2$, but I think it is interesting to ask the more general question.

Here is an example: if $F$ is q quadratic field, then the 2-torsion in the class group $C_F$ is generated by primes dividing the discriminant (see for example the answer to this question). In particular the order of the 2-torsion is an $O(D_F^\delta)$ for any $\delta > 0$.

I would be interested in similar bounds for fields of higher degree, in particular for non-real cubic fields. (In this case there seems to be a relation between the 2-torsion of the class group and elliptic curves but I am not competent to exploit it).

There are also much better bound "on average" for the 2-torsion of class groups of cubic fields: Manjul Bhargava has proven that when cubic fields are ordered by discriminant the mean order for the 2-torsion tends to a constant which is equal to 1.25 in the case of non-real fields. (I read about that and more in this preprint.)

Answer 1

Ellenberg and Venkatesh prove a number of bounds for the $\ell$-torsion in class groups in their paper Reflection principles and bounds for class group torsion.

They show, for instance, that if $\ell$ is a positive integer and $K$ is a number field of degree $d$ with class group $\mathrm{Cl}_K$ and discriminant $\mathrm{disc}(K)$ then under the assumption of GRH one has the bound

$$\#\mathrm{Cl}_K[\ell]\ll_{d,\epsilon} \mathrm{disc}(K)^{1/2-\frac{1}{2\ell(d-1)}+\epsilon}.$$

EDIT - I just noticed a preprint of Ellenberg, Pierce, and Matchett Wood which obtains further bounds for class group torsion. In their very nice introduction they explicitly mention that while a bound of the shape you were looking for ($\#\mathrm{Cl}_K[\ell]\ll \mathrm{disc}(K)^\epsilon$ for every $\epsilon>0$) is conjectured, it has been quite difficult to even improve upon the trivial bound $\#\mathrm{Cl}_K[\ell]\ll \mathrm{disc}(K)^{1/2+\epsilon}$.

Answer 2

I'm happy to announce a new result of the shape you ask for: if $F$ is a cubic field (of any signature) then the size of the 2-torsion in its class group is bounded above by $O(D_F^{0.2785})$. The same is also true if $F$ is a number field of any degree $n$, but in this case one has to replace $0.2785$ with a function of $n$ that rapidly tends to $1/2$.

This is joint work of Manjul Bhargava, Arul Shankar, Takashi Taniguchi, Jacob Tsimerman, Yongqiang Zhao, and myself. (We did this at an AIM mini-workshop; I highly recommend these to anyone who has the chance!)

I regret that the paper is in a rather rough state of preparation, and so I don't have a preprint to share. Once we're done, I'd be happy to e-mail a copy to you or to anyone else who leaves their contact information in the comments.

Answer 3

As already noted, estimates of the form $O(D_F^\delta)$ for every $\delta>0$ are conjectured, but we are far from a proof in even the easiest non-genus field case, which is probably 3-torsion in quadratic fields. The first non-trivial bounds are actually only about a decade old. Such bounds were proved independently by Pierce and by Helfgott and Venkatesh:

• Pierce, Lillian B., The 3-part of class numbers of quadratic fields, J. London Math. Soc. (2) 71 (2005), no. 3, 579–598. MR2132372
• Pierce, Lillian B., A bound for the 3-part of class numbers of quadratic fields by means of the square sieve. Forum Math. 18 (2006), no. 4, 677–698 MR2254390
• Helfgott, H. A., Venkatesh, A., Integral points on elliptic curves and 3-torsion in class groups. J. Amer. Math. Soc. 19 (2006), no. 3, 527–550 MR2220098