In the next few posts, I want to discuss a problem that came up when I wrote a paper with Barry Mazur. We had a few observations and remarks that we discussed as part of a possible sequel but which we never wrote up(*); mostly because we never could quite prove what we wanted to prove. But some of those remarks might be worth sharing.
The basic problem is as follows. Let E/Q be a number field of signature (r,s). Let p be a prime that splits completely in E (this is not strictly necessary, but it makes things cleaner). Let S be a set of primes above p. If S includes all the primes above p, then the Leopoldt Conjecture for E and p is the statement that
\(r_S:=\mathrm{dim}_{\mathbf{Q}_p} \mathrm{Gal}(E^S/E)^{\mathrm{ab}} \otimes \mathbf{Q} = 1+s.\)
The question is then to predict what happens when S is a strict subset of the primes above p. This leads to the following minimalist definition:
Definition: The field E is rigid at p if
$latex r_S:=\mathrm{dim}_{\mathbf{Q}_p} \mathrm{Gal}(E^S/E)^{\mathrm{ab}} \otimes \mathbf{Q}
= \begin{cases} \#S – (r+s-1), & \#S \ge r + s – 1,
\\ 0, & \text{otherwise}. \end{cases}$
Note that, for any field E, the right hand side is always a lower bound. So rigid pairs (E,p) are those which have no “unexpected” Z_p-extensions. If E is totally real, the Leopoldt Conjecture at p is equivalent to E being rigid. However, one does not predict that all fields E are rigid. The following is elementary:
Proposition If E is a totally imaginary CM field, then complex conjugation acts naturally on the set S. There are inequalities \(r_S \ge [E^+:\mathbf{Q}] + 1\) if S consists of all primes above p, and
\(r_S \ge \frac{1}{2} \# (S \cap c S)\)
otherwise. If Leopoldt’s conjecture holds, then these inequalities are equalities.
It follows that if E is a CM field of degree at least 4, then E is not rigid for any prime p, because when S consists of two primes conjugate to each other under complex conjugation, then
\(r_S \ge 1 > 2 – (r+s-1) = 2 – s.\)
The “extra” extensions are coming from algebraic Hecke characters. Our expectation is that this is the only reason for a pair (E,p) to be rigid. For example:
Conjecture: Suppose that E does not contain a totally imaginary CM extension F of degree at least 4. Then (E,p) is rigid for any prime p that splits completely in E.
(When I say conjecture here, I really mean a guess; it could be false for trivial reasons.) Naturally these conjectures are hard to prove, since they imply Leopoldt’s Conjecture. Even if one assumes Leopoldt’s Conjecture, this conjecture still seems tricky. It makes sense, however, to see what can be proven under further “genericity” hypotheses on the image of the global units inside the local units. To this end, let me recall the Strong Leopoldt Conjecture which Barry and I formulated our original paper. Let F/Q be the splitting field of E/Q. Let G be the Galois group of F/Q. There is a G-equivariant map
\(\mathcal{O}^{\times}_F \otimes \mathbf{Q}_p \rightarrow \prod_{v|p} \mathcal{O}^{\times}_{F,v} \otimes \mathbf{Q}_p.\)
The right hand side is isomorphic as a G-module to \(\mathbf{Q}_p[G]\). However, more is true; for a fixed prime v|p, there is an isomophism
\(\mathbf{Q}_p[G] = \mathbf{Q}[G] \otimes \mathbf{Q}_p\)
which is well-defined up to a scalar in \(\mathbf{Q}_p\) coming from a choice of p-adic logarithm for the given place at p. It makes sense to talk about a rational subspace V of the right hand side, namely, a space of the form \(V = V_{\mathbf{Q}} \otimes \mathbf{Q}_p\) for some \(V_{\mathbf{Q}} \subset \mathbf{Q}[G].\) The strong Leopoldt conjecture asserts that the intersection of the global units wich such a rational subspace is as small as it can possibly be subject to the constraints of the G-action on both V and the units, together with Leopoldt’s conjecture that the map from the units tensor \(\mathbf{Q}_p\) is injective. Let H = \Gal(F/E). By inflation-restriction, there is an isomorphism
\(H^1_S(E,\mathbf{Q}_p) = H^1_T(F,\mathbf{Q}_p)^{H},\)
where the subscript denotes classes “unramified outside S,” and where T denotes the set of primes in F above S. By class field theory, this may be identified with the H-invariants of the cokernel of the map
\(\mathcal{O}^{\times}_F \otimes \mathbf{Q}_p \rightarrow \prod_{T} \mathcal{O}^{\times}_{F,v} \otimes \mathbf{Q}_p.\)
The cokernel is larger than expected if and only if the kernel is bigger than expected. In particular, \(r_S = \mathrm{dim} H^1_S(E,\mathbf{Q}_p)\) is bigger than expected only if
\(\left( \mathcal{O}^{\times}_F \otimes \mathbf{Q}_p \bigcap \prod_{\neg T} \mathcal{O}^{\times}_{F,v} \otimes \mathbf{Q}_p \right)^{H}\)
is bigger than expected. Note that the product over any subset T of primes in the right hand side is a rational subspace. Certainly the Strong Leopoldt Conjecture determines the dimension of the intersection \(U \cap V\) of the unit group with a rational subspace. What is slightly less clear is that the intersection \((U \cap V)^{H}\) for any subgroup \(H\) is also determined by the strong Leopoldt Conjecture, but this is true (and we prove it). As a consequence, one has:
Lemma: Assuming the Strong Leopoldt Conjecture, the dimension \(r_S\) depends only on G, H, and S.
This “reduces” the computation of r_S to an intersection problem in a certain Grassmannian. But this is a computation we were never really able to do!
This is the problem: One knows very well the structure of the unit group of F as a G-module. So to compute the relevant intersections, one only has to compute the intersection with a “generic” rational subspace. Paradoxically, it seems very difficult in general to give explicit examples of rational subspaces which are generic enough to obtain the correct minimal value. So while, for formal reasons, almost any rational subspace will do, none of the nice subspaces which allow us to compute the intersection tend to be good enough.
Instead, to compute these intersections, we somewhat perversely look at actual number fields and their unit groups. This seems like a bad idea, since even verifying Leopoldt for a particular K and p is not so easy to do. So instead, we start with a totally real number field K of a certain form. Then, under the assumption of Leopolodt’s conjecture we can (non-constructively) find subspaces of rational subspaces V which provably minimize various intersections \(\mathrm{dim}(W \cap V)\) for various unit-like submodules W. We then deform the field K to other fields L of different signature, and use this construction (as well as the Strong Leopoldt Conjecture) to make deductions about L. In the next post, we explain how this led Barry and me to a proof of the following:
Theorem: Let E/Q be a degree n field with whose Galois closure F has Galois group G = S_n. Assume the Strong Leopoldt Conjecture. Then (E,p) is rigid for any prime p which splits completely in p.
I will explain the details next time. But to unwind the serpentine argument slightly, we do not prove the result by finding rational subspaces in \(\mathbf{Q}_p[G]\) whose intersection with the units of F has the a dimension which we can compute to be the expected value, but only rational subspaces whose dimension we can compute contingent on Leopoldt’s conjecture for some auxiliary totally real number field. In other words, we would like to compute the generic dimension of some intersection inside some G-Grassmannian, a problem which has nothing to do with number theory, and we compute it using Leopoldt’s conjecture. More next time!
(* never wrote up = actually written up in a pdf file on my computer somewhere)
