## Silas Johnson on weighted discriminants with mass formulas

My Ph.D. student Silas Johnson just posted his thesis paper to the arXiv, and it’s cool, and I’m going to blog about it!

How should you count number fields?  The most natural way is by discriminant; you count all degree-n number fields K with a given Galois group G in S_n and discriminant bounded in absolute value by B.  This gives you a value N_G(B) whose asymptotic behavior in B you might want to study.  The classical results and exciting new ones you’ve heard about — Davenport-Heilbron, Bhargava, and all that — generally concern counts of this kind.

But there are reasons to consider other kinds of counts.  I once had a bunch of undergrads investigate the behavior of N_3(X,Y), the number of cubic fields whose discriminant had squarefree part at most X and maximal square divisor at most Y.  This gives a more refined picture of the ramification behavior of the fields.  Asymptotics for this are still unknown!  (I would expect the main term to be on order $X Y^{1/2}$, but I don’t know what the secondary terms should be.)

One nice thing about the discriminant, though, is that it has a mass formula.  In brief:  a map f from Gal(Q_p) to S_n corresponds to a degree-n extension of Q_p, which has a discriminant (a power of p) which we call Disc(f).  Averaging Disc(f)^{-1} over all homomorphisms f gives you a polynomial in p^{-1}, which we call the local mass.  Now here’s the remarkable fact (shown by Bhargava, deriving from a formula of Serre) — there is a universal polynomial P(x) such that the local mass at p is equal to P(p^{-1}) for every P.  This is not hard to show for the tame primes p (you can see this point discussed in Silas’s paper or in the paper by Kedlaya I linked above) but that it holds for the wild primes is rather mysterious and strange.  And it certainly seems to ratify the idea that there’s something especially nice about the discriminant.  What’s more, this polynomial P is not just some random thing; it’s the product over p of P(p^{-1}) that gives the constant in Bhargava’s conjectural asymptotic for the number of number fields for degree n.

But here’s the thing.  If we replace G by a subgroup of S_n, there need not be a universal mass formula anymore.  Kedlaya (and Daniel Gulotta, in the appendix) show lots of examples.  The simplest example is the dihedral group of order 8.

All is not lost, though!  Wood showed in 2008 that you could fix this problem by replacing the discriminant of a D_4-extension with a different invariant.  Namely:  a D_4 quartic field M has a quadratic subextension L.  If you replace Disc(L/Q) with Disc(L/Q) times the norm to Q of Disc(L/M), you get a different invariant of M — an example of what Silas calls a “weighted discriminant” — and when you compute the local mass according to {\em this} invariant, you get a polynomial in p^{-1} again!

So maybe Wood’s modified discriminant, not the usual discriminant, is the “right” way to count dihedral quartics?  Does the product of her local masses give the right asymptotic for the number of D_4 extensions with Woodscriminant at most B?

It’s not at all clear to me how, if at all, you can cook up a modified discriminant for an arbitrary group G that has a universal mass formula.  What Silas shows is that having a mass formula is indeed special; when G is a p-group, there are only finitely many weighted discriminants that have one.  Silas thinks, and so do I, that this is actually true for every finite group G, and that some version of his approach will eventually prove this.  And he classifies all such weighted discriminants for groups of size up to 12; for any individual G, it’s a computation which can be made nicely algorithmic.  Very cool!

## Can the trace hear the shape of its field?

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.

## Modeling lambda-invariants by p-adic random matrices

The paper “Modeling λ-invariants by p-adic random matrices,” with Akshay Venkatesh and Sonal Jain, just got accepted by Comm. Pure. Appl.  Math. But I forgot to blog about it when we finished it!  (I was a little busy at the time with the change in my personal circumstances.)

Anyway, here’s the idea.  As I’ve already discussed here, one heuristic for the Cohen-Lenstra conjectures about the p-rank of the class group of a random quadratic imaginary field K is to view this p-part as the cokernel of g-1, where g is a random generalized symplectic matrix over Z_p.  In the new paper, we apply the same philosophy to the variation of the Iwasawa p-adic λ-invariant.

The p-adic λ-invariant of a number field K is closely related to the p-rank of the class group of K; in fact, Iwasawa theory more or less gets started from the theorem that the p-rank of the class group of $K(\zeta_{p^n})$ is

$\lambda n + \mu p^n + \nu$

for some constants $\lambda, \mu, \nu$ when n is large enough, with $\mu$ expected to be 0 (and proved to be 0 when K is quadratic.)  On the p-adic L-function side, the λ-invariant is (thanks to the main conjecture) related to the order of vanishing of a p-adic L-function.  On the function field side, the whole story is told by the action of Frobenius on the p-torsion of the Jacobian of a curve, which is specified by some generalized symplectic matrix g over F_p.  The p-torsion in the class group is the dimension of the fixed space of g, while the λ-invariant is the dimension of the generalized 1-eigenspace of g, which might be larger.  It’s also in a sense more natural, depending only on the characteristic polynomial of g (which is exactly what the L-function keeps track of.)

So in the paper we do two things.  On the one hand, we study the dimension of the generalized 1-eigenspace of a random generalized symplectic matrix, and from this we derive the following conjecture: for each p > 2 and r >= 0,  the probability that a random quadratic imaginary field K has p-adic λ-invariant r is

$p^{-r} \prod_{t > r} (1-p^{-t})$.

Note that this decreases like $p^{-r}$ with r, while the p-rank of the class group is supposed to be r with probability more like $p^{-r^2}$.  So large λ-invariants should be substantially more common than large p-ranks.

The second part of the paper tests this conjecture numerically, and finds fairly good agreement with the data. A novelty here is that we compute p-adic  λ-invariants of K for small p and large disc(K); previous computational work has held K fixed and considered large p.  It turns out that you can do these computations reasonably efficiently by interpolation; you can compute special values L(s,chi_K) transcendentally for many s; given a bunch of these values, determined to a certain p-adic precision, you can compute the initial coefficients of the p-adic L-function with some controlled p-adic precision as well, and, in particular, you can provably locate the first coefficient which is nonzero mod p.  The location of this coefficient is precisely the λ-invariant.  This method shows that, indeed, large λ-invariants do pop up!  For instance, the 3-adic λ-invariant of $Q(\sqrt{-956238})$ is 14, which I think is a record.

Some questions still floating around:

• Should one expect an upper bound $\lambda \ll_\epsilon D_K^\epsilon$ for each odd p?  Certainly such a bound is widely expected for the p-rank of the class group.
• In the experiments we did, the convergence to the conjectural asymptotic appears to be from below.  For the 3-ranks of class groups of quadratic imaginary fields, this convergence from below was conjectured by Roberts to be explained by a secondary main term with negative coefficient.  Roberts’ conjecture was proved this year — twice!  Bhargava, Shankar, and Tsimerman gave a proof along the lines of Bhargava’s earlier work (involving thoughful decompositions of fundamental domains into manageable regions, and counting lattice points therein) and Thorne and Taniguchi have a proof along more analytic lines, using the Shintani zeta function.  Anyway, one might ask (prematurely, since I have no idea how to prove the main term correct!) whether the apparent convergence from below for the statistics of the λ-invariant is also explained by some kind of negative secondary term.

## The different does not have a canonical square root

Just wanted to draw attention to this very nice exchange on Math Overflow.   Matt Emerton remarks that the different of a number field is always a square in the ideal class group, and asks:  is there a canonical square root of the ideal class of the different?

What grabs me about this question is that the word “canonical” is a very hard one to define precisely.   Joe Harris used to give a lecture called, “The only canonical divisor is the canonical divisor.”  The difficulty around the word “canonical” is what gives the title its piquancy.

Usually we tell students that something is “canonical” if it is “defined without making any arbitrary choices.”  But this seems to place a lot of weight on the non-mathematical word “arbitrary.”

Here’s one way to go:  you can say a construction is canonical if it is invariant under automorphisms.  For instance, the abelianization of a group is a canonical construction; if f: G_1 -> G_2 is an isomorphism, then f induces an isomorphism between the abelianizations.

It is in this sense that MathOverflow user “Frictionless Jellyfish” gives a nice proof that there is no canonical square root of the different; the slick cnidarian exhibits a Galois extension K/Q, with Galois group G = Z/4Z, such that the ideal class of the different of K has a square root (as it must) but none of its square roots are fixed by the action of G (as they would have to be, in order to be called “canonical.”)  The different itself is canonical and as such is fixed by G.

But this doesn’t seem to me to capture the whole sense of the word.  After all, in many contexts there are no automorphisms!  (E.G. in the Joe Harris lecture, “canonical” means something a bit different.)

Here’s a sample question that bothers me.  Ever since Gauss we’ve known that there’s a bijection between the set of proper isomorphism classes of primitive positive definite binary quadratic forms of discriminant d and the ideal class group of a quadratic imaginary field.

Do you think this bijection is “canonical” or not?  Why?