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.)
Best Answer
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 $\#\mathrm{Cl}_K[\ell]\ll \mathrm{disc}(K)^{1/2+\epsilon}$.