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.