3 The valuation function \(s\)
3.1 General properties
For \(a \in \mathbb {Z}\) and a maximal ideal \(\mathfrak {Q}\) of \(\mathcal{O}_L\), let \(s_{\mathfrak {Q}}(a) \in \mathbb {N}\) be the multiplicity of \(\mathfrak {Q}\) in the ideal \((g_a)\). We write \(s(a) = s_{\mathfrak {P}}(a) = v_{\mathfrak {P}}(g_a)\).
If \(a \equiv b \pmod{q-1}\), then \(s(a) = s(b)\).
Indeed \(g_a = g_b\) by 18.
If \((q - 1) \mid a\), then \(s(a) = 0\).
By 19, \(g_a = -1\) is a unit, so \((g_a) = \mathcal{O}_L\).
If \((q-1) \nmid (a+b)\), then \(s(a) + s(b) = s(a+b) + v_{\mathfrak {P}}(J(a, b))\).
For all \(a, b \in \mathbb {Z}\), \(s(a + b) \leq s(a) + s(b)\).
If \((q-1) \nmid a\), then \(s(a) + s(-a) = f \cdot e(\mathfrak {P}\mid p)\).
Take valuations in 22: the factor \(\omega ^{-a}(-1)\) is a root of unity, hence a unit, and \(v_{\mathfrak {P}}(p^f) = f \cdot e(\mathfrak {P}\mid p)\).
Assume \((q-1) \mid k\). Then
We have
Let \(\mathfrak {Q}\) be a maximal ideal of \(\mathcal{O}_L\) not lying above \(p\). Then \(s_{\mathfrak {Q}}(a) = 0\) for all \(a \in \mathbb {Z}\).
For all \(\sigma \in \mathrm{Gal}(L/F)\) and \(a \in \mathbb {Z}\), \(s_{\sigma (\mathfrak {P})}(a) = s_{\mathfrak {P}}(a \cdot \mathfrak {n}(\sigma ^{-1}))\).
The multiplicity of \(\sigma (\mathfrak {P})\) in \((g_a)\) is the multiplicity of \(\mathfrak {P}\) in \((\sigma ^{-1}(g_a))\), and \(\sigma ^{-1}(g_a) = g_{a \mathfrak {n}(\sigma ^{-1})}\) by 30.
3.2 Upper bound by the sum of digits
We have \(s(1) = 1\).
For all \(a \in \mathbb {N}\), \(s(a) \leq a\).
Let \(a \in \mathbb {N}\) and let \(a = a_0 + a_1 p + \cdots + a_{t-1} p^{t-1}\) be its \(p\)-adic expansion. Then
3.3 The digit formula
Let \(0 \leq a \leq q - 2\) and let \(a = a_0 + a_1 p + \cdots + a_{f-1} p^{f-1}\), \(0 \leq a_i \leq p - 1\), be the \(p\)-adic expansion of \(a\). Then
By 105, \(e(\mathfrak {P}\mid p) = p - 1\), so 39 gives \(\sum _{a=0}^{q-2} s(a) = (q-2)f(p-1)/2\). A direct count shows that the sum of the digit sums of \(0, \ldots , q-2\) has the same value. Since each term satisfies \(s(a) \leq a_0 + \cdots + a_{f-1}\) by 44, all these inequalities are equalities.
Let \(b {\gt} 1\), \(l \geq 0\) and \(0 \leq x {\lt} b^l - 1\). Then
where \(S_b(x)\) is the sum of the base \(b\) digits of \(x\).
Modulo \(b^l - 1\), multiplication by \(b\) cyclically rotates the \(l\) base \(b\) digits of \(x\). Hence each digit of \(x\) occurs exactly once in each position in the sum, which is therefore \(S_b(x)(1 + b + \cdots + b^{l-1})\).
Let \(m, d\) be positive integers with \(q - 1 = md\), and let \(a \in \mathbb {N}\) with \(da \leq q - 2\). Then