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 is

for some constants when n is large enough, with 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

.

Note that this decreases like with r, while the p-rank of the class group is supposed to be r with probability more like .  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 is 14, which I think is a record.

Some questions still floating around:

• Should one expect an upper bound 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?