New paper up with Lillian Pierce and Melanie Matchett Wood!
Here’s the deal. We know a number field K of discriminant D_K has class group of size bounded above by roughly D_K^{1/2}. On the other hand, if we fix a prime l, the l-torsion in the class group ought to be a lot smaller. Conjectures of Cohen-Lenstra type predict that the average size of the l-torsion in the class group of D_K, as K ranges over a “reasonable family” of algebraic number fields, should be constant. Very seldom do we actually know anything like this; we just have sporadic special cases, like the Davenport-Heilbronn theorem, which tells us that the 3-torsion in the class group of a random quadratic field is indeed constant on average.
But even though we don’t know what’s true on average, why shouldn’t we go ahead and speculate on what’s true universally? It’s too much to ask that Cl(K)[l] literally be bounded as K varies (at least if you believe even the most modest version of Cohen-Lenstra, which predicts that any value of dim Cl(D_K)[l] appears for a positive proportion of quadratic fields K) but people do think it’s small:
Conjecture: |Cl(K)[l]| < D_K^ε.
Even beating the trivial bound
|Cl(K)[l]| < |Cl(K)| < D_K^{1/2 + ε}
is not easy. Lillian was the first to do it for 3-torsion in quadratic fields. Later, Helfgott-Venkatesh and Venkatesh and I sharpened those bounds. I hear from Frank Thorne that he, Bhargava, Shankar, Tsimerman and Zhao have a nontrivial bound on 2-torsion for the class group of number fields of any degree.
In the new paper with Pierce and Wood, we prove nontrivial bounds for the average size of the l-torsion in the class group of K, where l is any integer, and K is a random number field of degree at most 5. These bounds match the conditional bounds Akshay and I get on GRH. The point, briefly, is this. To make our argument work, Akshay and I needed GRH in order to guarantee the existence of a lot of small rational primes which split in K. (In a few cases, like 3-torsion of quadratic fields, we used a “Scholz reflection trick” to get around this necessity.) At the time, there was no way to guarantee small split primes unconditionally, even on average. But thanks to the developments of the last decade, we now know a lot more about how to count number fields of small degree, even if we want to do something delicate like keep track of local conditions. So, for instance, not only can one count quartic fields of discriminant < X, we can count fields which have specified decomposition at any specified finite set of rational primes. This turns out to be enough — as long as you are super-careful with error terms! — to allow us to show, unconditionally, that most number fields of discriminant < D have enough small split primes to make the bound on l-torsion go. Hopefully, the care we took here to get counts with explicit error terms for number fields subject to local conditions will be useful for other applications too.
Quomodocumque 