There are no unramified abelian extensions of Q (almost)

In my class on modularity, I decided to explain what Wiles’ argument (in the minimal case) would look like for \(\mathrm{GL}(1)/F\). There are two ways one can go with this. On the one hand, one can try to prove (say) Kronecker-Weber using Selmer groups, but avoiding any kind of circularity (by not assuming class field theory). On the other hand, one can allow oneself to be completely circular in an effort to concentrate on the technical details of Wiles’ arguments. This post concerns the latter, and we “prove” the following:

Theorem: Let \(F\) be a number field which does not contain \(\zeta_p\). Then the Galois group of the maximal abelian extension of \(F\) unramified everywhere is isomorphic to the \(p\)-part of the class group.

To prove this, we will (only) assume the following:

The first three assumptions for \(F = \mathbf{Q}\) are equivalent to giving oneself the cyclotomic extensions and understanding their ramification properties. The last assumption, of course, contains every part of global class field theory (making the argument circular).

Let \(\Gamma_{\emptyset}\) denote the Galois group of the maximal pro-\(p\) abelian unramified extension of \(F\).
Let \(\Gamma_{Q_N}\) denote the corresponding group where ramification is allowed at some set of primes \(Q_N\) not containing \(p\), where one also insists that the order of inertia at primes in \(Q_N\) is at most \(p^N\). Formally, we have the universal deformation rings

\(R_{\emptyset} = \mathbf{Z}_p[[\Gamma_{\emptyset}]], \qquad R_{Q_N} = \mathbf{Z}_p[[\Gamma_{Q_N}]],\)

and we also have the universal “modular” deformation rings

\(\mathbb{T}_{\emptyset} = \mathbf{Z}_p[[(F^{\times}\backslash \mathbb{A}^{\times}_{F}/U)^F]], \qquad \mathbb{T}_{Q_N} = \mathbf{Z}_p[[(F^{\times}\backslash \mathbb{A}^{\times}_{F}/U_Q)^F]].\)

Here $M^F$ denotes the biggest finite quotient of $M$, \(U\) is the obvious maximal open compact, and \(U_Q\) is the variant of \(U\) such that \(U_v = \mathcal{O}^{\times}_v\) is replaced by \(U_{Q_N,v} = \mathcal{O}^{\times p^N}_v\) for \(v \in Q_N\). The half of global class field theory we are assuming gives us a compatible diagram of maps \(R_{Q_N} \rightarrow \mathbb{T}_{Q_N}\) and \(R_{\emptyset} \rightarrow \mathbb{T}_{\emptyset}\). The Wiles-Greenberg formula gives us an equality:

\(\dim |H^1_{\emptyset}(F,\mathbf{F}_p)| – \dim |H^1_{\emptyset^*}(F,\mathbf{F}_p(1))|= – (r_1 + r_2 – 1),\)

where for this computation we use that \(\zeta_p \notin F\). In order to annihilate the dual Selmer group, we need to annihilate classes in \(H^1(F,\mu_p)\), which come from extensions \(F(\zeta_p,\sqrt[p]{\alpha})\). We can do this in the usual way, but we have to assume (again) that \(\zeta_p \notin F\), since otherwise one cannot annihilate the class defined over \(F(\zeta_{p^2}) = F(\zeta_p,\sqrt[p]{\zeta_p})\) using a prime \(q \equiv 1 \mod p^2\). We see that we can annihilate the dual Selmer group with \(q:=|Q_N| = \dim H^1_{\emptyset} + (r_1 + r_2 – 1)\) primes. What are the auxilary rings \(S_N\) here? As rings, they are

\(S_N = \mathbf{Z}_p[(\mathbf{Z}/p^N \mathbf{Z})^q]\)

the action on \(R_{Q_N}\) is via the inertia group at the auxiliary primes. To make this work, one needs local class field theory; this shows that inertia at \(q\) is acting via \(\mathcal{O}^{\times}_q/\mathcal{O}^{\times p^N}_q\). The action of \(S_N\) on \(\mathbb{T}_{Q_N}\) is given by the structure of \(\mathbb{T}_{Q_N}\) as a module over \(\mathbf{Z}_p[U/U_Q] \simeq S_N\). The compatibility of these actions is given by the compatibility of local and global class field theory. Moreover, if \(\mathfrak{a}_N\) is the augmentation ideal of \(S_N\), then \(R_{Q_N}/\mathfrak{a}_N = R_{\emptyset}\) by definition, and \(\mathbb{T}_{Q_N}/\mathfrak{a}_N = \mathbb{T}_{\emptyset}\) by construction. Thus, in the usual way, one ends up with a map \(R_{\infty} \rightarrow \mathbb{T}_{\infty}\) where:

The final thing to understand is the structure of \(\mathbb{T}_{\infty}\) as a module over \(S_{\infty}\). At level \(Q_N\), the annihilator in \(S_N \simeq \mathbf{Z}_p[U/U_Q]\) of \(\mathbb{T}_{Q_N}\) is given by the image of the global units. By Dirichlet’s Theorem, this is generated by at most \(r_1 + r_2 – 1\) generators (assuming again that \(\zeta_p \notin F\)). By patching, it follows (in the limit) that \(\mathbb{T}_{\infty}\) has co-dimension at most \(r_1 + r_2 – 1\), and thus (from dimension considerations) that \(\mathbb{T}_{\infty} = R_{\infty}\), and then (after taking the quotient by \(\mathfrak{a}_{\infty}\)) that \(R_{\emptyset} \simeq \mathbb{T}_{\emptyset}\), which proves that \(\Gamma_{\emptyset}\) is the \(p\)-part of the class group. Note that (as expected) when gluing, we need to take into account all the (finitely many) possible \(R/\mathfrak{m}^N\), \(\mathbb{T}/\mathfrak{m}^N\), the possible maps from the global units to \(S_N\), &. &.

edit: In order to see the “circularity” more clearly, one may compute the Selmer groups directly. The group \(H^1_{\emptyset}(F,\mathbf{F}_p)\) is equal to \(\Gamma_{\emptyset}/p\), by definition. On the other hand, the group \(H^1_{\emptyset^*}(F,\mu_p)\) by the Kummer sequence is equal to \(\mathcal{O}^{\times}/\mathcal{O}^{\times p} \oplus \mathrm{Pic}^0(\mathrm{Spec}(\mathcal{O}_F))[p]\), and thus the Greenberg-Wiles formula is equivalent to the equality:

\(|\mathrm{Pic}^0(\mathrm{Spec}(\mathcal{O}_F))[p]| = |\Gamma_{\emptyset}/p \Gamma_{\emptyset}|\)

or equivalently the claim that the maximal exponent \(p\)-quotient of the class group captures all exponent \(p\)-unramified extensions. (I guess this is very very slightly weaker than \(\Gamma_{\emptyset} \otimes \mathbf{Z}_p \simeq \mathrm{Pic}^0(\mathrm{Spec}(\mathcal{O}_F)) \otimes \mathbf{Z}_p\)).