For \(K\) a finite extension of \(\QQ_p\text{,}\) the local reciprocity map defines an isomorphism of \(\Gal(\overline{K}/K)^{\ab}\) with the profinite completion of \(K\text{.}\) The natural filtration on the unit group \(\gotho_K^\times\) thus defines a filtration on \(\Gal(\overline{K}/K)^{\ab}\text{;}\) but which one? It turns out that the answer is related to a natural filtration on the entire group \(\Gal(\overline{K}/K)\text{;}\) we give Hadamard’s description of this.
SubsectionThe lower numbering filtration
Remark4.4.1.
Recall that for any extension \(L/K\) of finite extensions of \(\QQ_p\text{,}\) the ring \(\gotho_L\) is a monogenic extension of \(\gotho_K\text{:}\) there exists an element \(\alpha \in \gotho_L\) such that \(\gotho_L = \gotho_K[\alpha]\text{,}\) meaning that the \(\gotho_K\)-linear homomorphism \(\gotho_K[x] \to \gotho_L\) taking \(x\) to \(\alpha\) is an isomorphism. (See [46], II.6, Proposition 12 or [37], Lemma II.10.4.)
Lemma4.4.2.
Let \(L/K\) be a Galois extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) Let \(v_L\) be the valuation on \(L\) and choose a uniformizer \(\pi_L\) of \(L\) (so that \(v_L(\pi_L) = 1\)). Choose \(\alpha \in \gotho_L\) such that \(\gotho_L = \gotho_K[\alpha]\text{.}\) For every \(g \in G\) and every integer \(i \geq -1\text{,}\) the following conditions are equivalent.
The action of \(g\) on the ring \(\gotho_L/\pi_L^{i+1}\) is trivial.
For all \(x \in \gotho_L\text{,}\)\(v_L(x^g - x) \geq i+1\text{.}\)
We have \(v_L(\alpha^g - \alpha) \geq i+1\text{.}\)
Proof.
The first two conditions are equivalent more or less by definition. They both immediately imply the third condition; conversely, the third condition implies the others because \(g\) fixes \(\gotho_K\) and \(\gotho_L = \gotho_K[\alpha]\text{.}\)
Definition4.4.3.
Let \(L/K\) be a Galois extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) For each integer \(i \geq -1\text{,}\) let \(G_i\) be the set of \(g \in G\) satisfying the equivalent conditions of Lemma 4.4.2. The \(G_i\) form a decreasing sequence of subgroups of \(G\text{;}\) these together form the lower numbering ramification filtration on \(G\text{.}\) In particular, \(G_{-1} = G\) and \(G_0\) equals the inertia subgroup of \(G\text{.}\)
For convenience later, we extend the definition of the filtration \(G_i\) to arbitrary real values \(i \geq -1\) by setting \(G_i = G_{\lceil i \rceil}.\)
From the definition, we see that the formation of the lower numbering filtration is compatible with subgroups: if \(H = \Gal(L/M)\) is a subgroup of \(G\text{,}\) then \(H_i = H \cap G_i\) for all \(i \geq -1\text{.}\) However, it is not at all clear what happens when we pass from \(G\) to a quotient.
Lemma4.4.4.
With notation as in Lemma 4.4.2, for \(i \geq 0\text{,}\) an element \(g \in G_0\) belongs to \(G_i\) if and only if \(\pi_L^g/\pi_L \equiv 1 \pmod{\pi_L^i}\text{.}\)
Proof.
Reduce to the case where \(L/K\) is totally ramified; we may then deduce the claim directly from Lemma 4.4.2. See also [46], IV.2, Proposition 5.
Definition4.4.5.
For \(i \geq 0\text{,}\) let \(U_L^i\) be the subgroup of \(\gotho_L^*\) consisting of elements \(\alpha\) for which \(v_L(\alpha - 1) \geq i\text{.}\) The group \(U_L^0/U_L^1\) is naturally isomorphic to the group of units of the residue field \(\gotho_L/\pi_L\text{.}\) For \(i > 0\text{,}\) the group \(U_L^i/U_L^{i+1}\) carries the structure of a one-dimensional vector space over \(\gotho_L/\pi_L\text{;}\) for any choice of the uniformizer \(\pi_L\) we may use the class of \(\pi_L^i\) as the basis element, but there is no distinguished choice without this breaking of symmetry.
By Lemma 4.4.4, for \(i \geq 0\) we may view \(G_i\) as the maximal subgroup of \(G\) carrying \(U_L^0\) into itself. In particular, the quotient \(G_i/G_{i+1}\) is naturally isomorphic to a subgroup of \(U_L^i/U_L^{i+1}\text{.}\)
This gives us the following structural properties of \(G\text{.}\) First, the group \(G_{-1}/G_0\) is isomorphic to the residue field extension, which is cyclic. Next, \(G_0/G_1\) is isomorphic to a subgroup of \(U_L^0/U_L^1\text{,}\) and so is cyclic of order prime to \(p\text{.}\) Finally, for \(i \geq 1\text{,}\)\(G_i/G_{i+1}\) is a subgroup of \(U_L^i/U_L^{i+1}\text{,}\) and so is an elementary abelian \(p\)-group. In particular, \(G\) is a solvable group, as noted in Remark 4.2.3.
SubsectionThe Herbrand functions
We now introduce Herbrand’s recipe to convert the lower numbering used in the definition of the ramification filtration into an upper numbering that behaves well with respct to passage to quotients.
Definition4.4.6.
Retain notation as in Definition 4.4.3. Define the function \(\varphi_{L/K}: [-1, \infty) \to [-1, \infty)\) by the formula
This function is continuous, piecewise linear, increasing, and concave, and satisfies \(\varphi_{L/K}(u) = u\) for \(u \in [-1,0]\text{.}\) Consequently, it admits an inverse \(\psi_{L/K}: [-1, \infty) \to [-1, \infty)\) which is continuous, piecewise linear, increasing, and convex.
We define the upper numbering on the ramification groups by the formula
Let \(L/K\) be a finite Galois extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) Let \(H\) be a normal subgroup of \(G\) with fixed field \(K'\text{.}\) For \(i \geq -1\text{,}\)\((G/H)_i = G_{\phi_{L/K'}(i)} H/H\text{.}\)
Let \(L/K\) be a finite Galois extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) Let \(H\) be a normal subgroup of \(G\) with fixed field \(K'\text{.}\) Then
Let \(L/K\) be a finite Galois extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) Let \(H\) be a normal subgroup of \(G\) with fixed field \(K'\text{.}\) For all \(i \geq -1\text{,}\)\((G/H)^i = G^i H/H\text{.}\)
Let \(L/K\) be a Galois extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) We define the breaks in the ramification filtration for the lower numbering (respectively, the upper numbering) as the values of \(i\) for which \(G_i \neq G_j\) for all \(j > i\) (resp. \(G^i \neq G^j\) for all \(j > i\)).
By definition, the breaks for the lower numbering are integers, while the breaks for the upper numbering are only guaranteed to be rational numbers. In fact, it is possible to exhibit examples where the breaks for the upper numbering are not integers (see Exercise 2 and Exercise 3). However, in the next section we will see that this cannot occur for abelian extensions.
SubsectionThe Hasse-Arf theorem
Theorem4.4.11.
Let \(L/K\) be an abelian extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) Then the breaks in the ramification filtration for the upper numbering are integers.
Consider the extension \(\QQ_p(\zeta_{p^n})/\QQ_p\text{.}\) One can compute directly (see Exercise 1) that the ramification breaks occur at \(1, \dots, n\text{.}\) This will also follow from the comparison with local reciprocity (Theorem 4.4.14).
Remark4.4.13.
The Hasse-Arf theorem is more general than we have stated here; it holds whenever \(L/K\) is a finite abelian extension of complete discretely valued fields in which the residue field extension is separable. That is, not only is there no restriction to characteristic 0, but the residue fields are not required to be finite.
At the same level of generality, one can use the Hasse-Arf theorem to deduce that the Artin conductor of a Galois representation is always integral. See [46], VI.2, Theorem 1.
Theorem4.4.14.
Let \(L/K\) be an abelian extension of finite extensions of \(\QQ_p\) with Galois group \(G\text{.}\) Let \(r_{L/K}: K^* / \Norm_{L/K} L^* \to G\) be the local reciprocity isomorphism. Then for each positive integer \(i\text{,}\) the inverse image of \(G^i\) in \(\gotho_K^*\) equals \(U_K^i\text{.}\)
Proof.
See [37], Theorem V.6.2. (This proof uses the Lubin-Tate construction.)
ExercisesExercises
1.
Compute the ramification breaks for the lower and upper numbering for the extension \(\QQ_p(\zeta_{p^n})/\QQ_p\) directly from the definitions (i.e., without using local reciprocity). In particular, you should find that the breaks for the upper numbering are \(1, \dots, n\text{.}\)
2.
Let \(K\) be the splitting field of the polynomial \(x^4 + 2x + 2\) over \(\QQ_2\text{.}\) Show that in the ramification filtration on \(\Gal(K/\QQ_2)\text{,}\) the largest break for the upper numbering occurs at \(4/3\text{.}\)
Hint.
This example is taken from the L-Functions and Modular Forms Database. Note that in this case the Galois group is \(S_4\text{.}\)
3.
Let \(G\) be the quaternion group of order \(8\text{;}\) that is, \(G = \{\pm 1, \pm i, \pm j, \pm k\}\text{.}\) Let \(C = \{\pm 1\}\) be the center of \(G\text{.}\) Suppose that \(L/K\) is a totally ramified Galois extension of finite extensions of \(\QQ_2\) satisfying \(\Gal(L/K) = G\) and \(G_4 = \{1\}\text{.}\) Show that
\begin{equation*}
G = G_0 = G_1, \qquad C = G_2 = G_3
\end{equation*}
and deduce that
\begin{equation*}
G^i = \begin{cases} G \amp i \leq 1 \\ C \amp 1 < i \leq \frac{3}{2} \\ \{1\} \amp i > \frac{3}{2}. \end{cases}
\end{equation*}
