[proofplan] We compute the upper numbering filtration $G^t(L_n/K)$ from the lower numbering via the Herbrand function $\eta_{L_n/K}$. First, we recall the lower ramification filtration: $G_s(L_n/K) = \operatorname{Gal}(L_n/L_k)$ for $q^{k-1} - 1 < s \leq q^k - 1$ (from the computation of the ramification breaks of $L_n/K$). We then integrate the index function $1/(G_0 : G_x)$ piecewise to obtain $\eta_{L_n/K}(s)$, invert this function to get $\psi_{L_n/K}(t)$, and apply $G^t = G_{\psi(t)}$ to recover the upper numbering. The result is that the upper numbering has integer jumps at $t = 0, 1, 2, \ldots, n-1$. [/proofplan] [step:Recall the lower ramification filtration of $L_n/K$] From the computation of lower ramification groups (using the [Herbrand's Theorem](/theorems/???) framework and the explicit computation of $i_{L_n/K}$), the lower numbering filtration is: \begin{align*} G_s(L_n/K) = \begin{cases} \operatorname{Gal}(L_n/K) & -1 \leq s \leq 0, \\ \operatorname{Gal}(L_n/L_1) & 0 < s \leq q - 1, \\ \operatorname{Gal}(L_n/L_2) & q - 1 < s \leq q^2 - 1, \\ \quad \vdots \\ \operatorname{Gal}(L_n/L_k) & q^{k-1} - 1 < s \leq q^k - 1, \\ \quad \vdots \\ \operatorname{Gal}(L_n/L_{n-1}) & q^{n-2} - 1 < s \leq q^{n-1} - 1, \\ \{1\} & s > q^{n-1} - 1. \end{cases} \end{align*} The subgroup orders are $|G_s| = |\operatorname{Gal}(L_n/L_k)| = [L_n : L_k] = q^{n-k}(q-1)/(q-1) \cdot (q-1)$... let us be precise. We have $|\operatorname{Gal}(L_n/K)| = q^{n-1}(q-1)$ and $|\operatorname{Gal}(L_n/L_k)| = [L_n : L_k]$. Since $[L_k : K] = q^{k-1}(q-1)$, we get $[L_n : L_k] = q^{n-1}(q-1) / (q^{k-1}(q-1)) = q^{n-k}$. In particular: - $|G_0| = |\operatorname{Gal}(L_n/K)| = q^{n-1}(q-1)$ (since $L_n/K$ is totally ramified, $G_0 = G_{-1}$). - For $0 < s \leq q - 1$: $|G_s| = |\operatorname{Gal}(L_n/L_1)| = q^{n-1}$. - For $q^{k-1} - 1 < s \leq q^k - 1$: $|G_s| = q^{n-k}$. [/step] [step:Compute the Herbrand function $\eta_{L_n/K}(s)$ by integrating the index function] The Herbrand function is defined by \begin{align*} \eta_{L_n/K}(s) = \int_0^s \frac{1}{(G_0 : G_x)} \, dx \quad \text{for } s \geq 0, \end{align*} extended by $\eta(s) = s$ for $-1 \leq s \leq 0$. The integrand $1/(G_0 : G_x)$ is a piecewise constant function of $x$. For $0 \leq x \leq q - 1$: $(G_0 : G_x) = |G_0|/|G_x| = q^{n-1}(q-1) / q^{n-1} = q - 1$, so $1/(G_0 : G_x) = 1/(q-1)$. For $q^{k-1} - 1 < x \leq q^k - 1$ (with $k \geq 2$): $(G_0 : G_x) = q^{n-1}(q-1) / q^{n-k} = q^{k-1}(q-1)$, so $1/(G_0 : G_x) = 1/(q^{k-1}(q-1))$. We integrate piecewise. For $0 \leq s \leq q - 1$: \begin{align*} \eta(s) = \int_0^s \frac{dx}{q-1} = \frac{s}{q-1}. \end{align*} At $s = q - 1$: $\eta(q-1) = 1$. For $q - 1 \leq s \leq q^2 - 1$ (i.e., $k = 2$): \begin{align*} \eta(s) = \eta(q-1) + \int_{q-1}^s \frac{dx}{q(q-1)} = 1 + \frac{s - (q-1)}{q(q-1)}. \end{align*} At $s = q^2 - 1$: $\eta(q^2 - 1) = 1 + \frac{q^2 - 1 - (q-1)}{q(q-1)} = 1 + \frac{q^2 - q}{q(q-1)} = 1 + \frac{q(q-1)}{q(q-1)} = 2$. In general, for $q^{k-1} - 1 \leq s \leq q^k - 1$: \begin{align*} \eta(s) = (k-1) + \frac{s - (q^{k-1} - 1)}{q^{k-1}(q-1)}. \end{align*} This is verified by induction: at $s = q^k - 1$, we get $\eta(q^k - 1) = (k-1) + \frac{q^k - 1 - q^{k-1} + 1}{q^{k-1}(q-1)} = (k-1) + \frac{q^{k-1}(q-1)}{q^{k-1}(q-1)} = k$. [guided] The Herbrand function $\eta$ is a piecewise linear, continuous, strictly increasing function that maps the lower numbering breaks to integer values. The computation amounts to integrating the step function $1/(G_0 : G_x)$ over each interval where $G_x$ is constant. The key observation is the index computation. On the interval $q^{k-1} - 1 < x \leq q^k - 1$, the lower ramification group is $G_x = \operatorname{Gal}(L_n/L_k)$ of order $q^{n-k}$, while $G_0 = \operatorname{Gal}(L_n/K)$ has order $q^{n-1}(q-1)$. The index is \begin{align*} (G_0 : G_x) = \frac{q^{n-1}(q-1)}{q^{n-k}} = q^{k-1}(q-1). \end{align*} The length of the interval is $q^k - 1 - (q^{k-1} - 1) = q^{k-1}(q-1)$. So the integral over this interval is $\frac{q^{k-1}(q-1)}{q^{k-1}(q-1)} = 1$. This is why $\eta$ increases by exactly $1$ on each interval: the length of the interval and the index conspire to give integer jumps. This "conspiracy" is a manifestation of the Hasse--Arf phenomenon. At the critical values: $\eta(0) = 0$, $\eta(q - 1) = 1$, $\eta(q^2 - 1) = 2$, ..., $\eta(q^k - 1) = k$, ..., $\eta(q^{n-1} - 1) = n - 1$. [/guided] [/step] [step:Invert $\eta$ to obtain $\psi_{L_n/K}(t)$ and compute the upper numbering] The function $\eta_{L_n/K}$ is continuous, piecewise linear, and strictly increasing, so it has a well-defined inverse $\psi_{L_n/K} = \eta_{L_n/K}^{-1}$. From the formula $\eta(s) = (k-1) + \frac{s - (q^{k-1} - 1)}{q^{k-1}(q-1)}$ for $q^{k-1} - 1 \leq s \leq q^k - 1$, the image is $k - 1 \leq \eta(s) \leq k$. Solving for $s$ in terms of $t = \eta(s)$: \begin{align*} s = \psi(t) = (q^{k-1} - 1) + q^{k-1}(q-1)(t - (k-1)) \quad \text{for } k-1 \leq t \leq k. \end{align*} In particular, $\psi(k-1) = q^{k-1} - 1$ and $\psi(k) = q^k - 1$, confirming consistency. Now apply the definition $G^t(L_n/K) = G_{\psi(t)}(L_n/K)$. For $k - 1 < t \leq k$ (with $1 \leq k \leq n - 1$): \begin{align*} \psi(t) \in (q^{k-1} - 1, q^k - 1], \end{align*} and on this interval the lower numbering gives $G_{\psi(t)} = \operatorname{Gal}(L_n/L_k)$. For $-1 \leq t \leq 0$: $\psi(t) = t$ (since $\eta(s) = s$ for $-1 \leq s \leq 0$), so $G^t = G_t = \operatorname{Gal}(L_n/K)$. For $t > n - 1$: $\psi(t) > q^{n-1} - 1$, and $G_s = \{1\}$ for $s > q^{n-1} - 1$, so $G^t = \{1\}$. Therefore: \begin{align*} G^t(L_n/K) = \begin{cases} \operatorname{Gal}(L_n/K) & -1 \leq t \leq 0, \\ \operatorname{Gal}(L_n/L_k) & k-1 < t \leq k, \; k = 1, \ldots, n-1, \\ \{1\} & t > n-1. \end{cases} \end{align*} Equivalently, for $0 \leq t \leq n - 1$, we have $G^t(L_n/K) = \operatorname{Gal}(L_n/L_{\lceil t \rceil})$ (setting $L_0 = K$), since $\lceil t \rceil = k$ for $k - 1 < t \leq k$. [guided] The passage from lower to upper numbering via the Herbrand function is a renumbering that "straightens out" the ramification breaks. In the lower numbering, the breaks occur at $q^k - 1$ (which grow exponentially). In the upper numbering, the breaks occur at the integers $0, 1, 2, \ldots, n - 1$. This is the content of the [Hasse--Arf Theorem](/theorems/???): for abelian extensions, the upper numbering breaks are integers. The Lubin--Tate extensions provide an explicit verification. The computation $\eta(q^k - 1) = k$ shows directly that the lower break at $q^k - 1$ maps to the upper break at $k$. The piecewise linear structure of $\eta$ means that $\psi = \eta^{-1}$ is also piecewise linear, mapping the interval $(k-1, k]$ in the upper numbering to the interval $(q^{k-1} - 1, q^k - 1]$ in the lower numbering. Since $G_s$ is constant on each interval of the lower numbering, the substitution $G^t = G_{\psi(t)}$ gives $G^t$ constant on each interval of the upper numbering. The advantage of the upper numbering is its compatibility with quotients (the [Compatibility of Upper Numbering with Quotients](/theorems/???) theorem): for a normal subgroup $H \trianglelefteq G$, the upper numbering satisfies $(G/H)^t = G^t H / H$. This property fails for the lower numbering and is the primary reason for introducing the Herbrand function. [/guided] [/step]