Tag Archives: number fields

“On l-torsion in class groups of number fields” (with L. Pierce, M.M. Wood)

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.

 

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Can the trace hear the shape of its field?

(A post about Guillermo Mantilla-Soler’s paper posted on the arXiv yesterday.)

The most natural arithmetic invariant of a number field K is its discriminant D_K, an integer congruent to either 0 or 1 mod 4 whose prime factors are precisely the primes where K/Q is ramified.  Oftentimes D_K is a squarefree, in which case it’s just the product of the ramified primes; even if not, the multiplicity of a prime factor of D_K can be described quite cleanly in terms of the p-inertia subgroup of Gal(K/Q).

The situation is especially handsome for quadratic field.  The discriminants of quadratic fields are just those integers congruent to 0 or 1 mod 4 which have no square factor larger than 4.  Better still, the discriminant specifies the field uniquely!  So in order to describe a quadratic field it suffices to write down a single integer.

Life gets worse in higher degree.  There can be lots of number fields with the same discriminant D.  For example:  if D is squarefree and K is a cubic field with discriminant D, then the Galois closure L of K is an unramified (Z/3Z)-extension of the quadratic field M with discriminant D.  So if the ideal class group of M has a lot of (Z/3Z) in it, there are going to be a lot of cubic fields with discriminant D!

Just how bad is this multiplicity?  It’s widely believed that, for every n, there are at most D^eps number fields of discriminant D.  But I think nobody has a good idea about how to prove this, even for n=3.

So it’s naturally interesting to ask whether there are other invariants which might uniquely specify a number field.  Here’s one natural candidate.  The ring of integers O_K is a free rank n Z-module, endowed with a natural quadratic form q(x,y) = Tr_{K/Q}(xy), called the trace form.  The discriminant of this trace form is, up to known factors, the discriminant of K.  So you can think of the isomorphism class of the trace form as a refinement of the discriminant.  The question is:  is it such a good refinement that it actually specifies the field?

My former student Guillermo Mantilla-Soler, now at UBC,  just posted a preprint offering the first real insight into this question, which he colorfully phrases “Can the trace hear the shape of its field?”  He shows that the answer to the original question is no:  for instance, he displays two non-isomorphic quintic fields of discriminant 34129 which have isomorphic trace form.  More generally, he gives a necessary condition which I would expect is satisfied by examples in every degree (though it might be hard to prove this.)

But in some sense this is a local issue; the fields in these examples are not totally real, so the trace forms aren’t definite, and so, as Guillermo observes, it suffices to show the forms lie in the same spinor genus.  In the definite case, it’s “harder” for quadratic forms to be isomorphic.  Are there two non-isomorphic totally real number fields with isomorphic trace forms?  Guillermo includes the results of a fairly large computer search which finds no examples in degree < 10 and discriminant < 10^9.

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