The integral of the root count #
Fix a totally ramified degree-n subextension L of a separable closure of K. A finite
disjoint cover of the uniformizers of L by balls of order larger than its discriminant
exponent counts every root of an Eisenstein polynomial exactly once. Integrating this count
over the Eisenstein region gives the sum of the measures of its coefficient fibres: the
polynomials having a root in each fixed ball.
Each fibre is measurable because a closed valuation ball is compact and projection of the closed evaluation-zero locus along that ball is closed. Consequently the decomposition holds for every measure on coefficient space, without Haar, normalization, regularity or separability assumptions on the polynomials. This isolates the integration step from the arithmetic calculation of the individual fibre volumes.
When the balls are centred at uniformizers and have order at least 2, the fibre of the
Eisenstein region over each ball is the image of the set of all integral coefficient vectors
whose polynomial has a root in that ball. The integral is therefore also a sum of measures of
such integral fibres, the sets on which the local root criterion at the centre and the volume
computation in power-basis coordinates operate.
Once those volumes are known, the integral is determined. Take balls of order r with 1 < r
and d < r, where d is the discriminant exponent of L. If the coefficient fibre over each
such ball centred at a uniformizer has measure q ^ (-(r + d)), the root-count integral is
q ^ (-(d + 1)) (1 - q⁻¹), Serre's value. The reason is that a partition of the uniformizers
into such balls has q ^ (r - 1) - q ^ (r - 2) members, and L / K has residue degree one, so
q is the residue cardinality of both K and L.
For a left-invariant measure normalized on the integer box, the local fibre criterion computes
each such fibre volume as q ^ (-(r + d)), so the root count of every L ∈ σ_K(n) integrates
over the Eisenstein region to q ^ (-(d + 1)) (1 - q⁻¹). Summed over σ_K(n), this gives
Serre's mass formula.
Main results #
IntermediateField.lintegral_rootCount_eq_sum_measure: the root-count integral is the sum of the measures of the coefficient fibres of a disjoint ball cover of the uniformizers.IntermediateField.lintegral_rootCount_eq_sum_measure_image: for a partition of the uniformizers into balls of order at least2, the root-count integral is the sum of the measures of the integral coefficient fibres.IntermediateField.lintegral_rootCount_eq_of_measure_eq: if every such fibre over a ball of orderrhas measureq ^ (-(r + d)), the root-count integral isq ^ (-(d + 1)) (1 - q⁻¹).IntermediateField.lintegral_rootCount: for a left-invariant measure normalized on the integer box, the root-count integral isq ^ (-(d + 1)) (1 - q⁻¹).
References #
- J.-P. Serre, Une « formule de masse » pour les extensions totalement ramifiées de degré donné d'un corps local, C. R. Acad. Sci. Paris Sér. A 286 (1978), 1031–1036, §3, Lemmas 2–3.
Integrate the root count in a counted extension by summing the measures of the coefficient fibres of any fixed finite disjoint ball cover of its uniformizers. The ball order must be larger than the discriminant exponent, so every occupied ball contains exactly one root.
Root-count integrals as sums of integral coefficient fibres. Let L ∈ σ_K(n), with its
canonical local-field structure, and let C index a partition of the uniformizers of L into
balls v_L(x - c) ≥ r whose order r exceeds both 1 and the discriminant exponent d(L).
Then the integral of L.rootCount over the Eisenstein region is the sum, over the balls, of the
measures of the sets of coefficient vectors of monic polynomials over 𝒪[K] having a root in
the ball. The bound 1 < r follows from d(L) < r unless n = 1, where d(L) = 0.
The root-count integral from the local fibre volumes. Let L ∈ σ_K(n), with discriminant
exponent d, and let 1 < r and d < r. Suppose that for every uniformizer c of L, the
coefficient vectors in the Eisenstein region whose polynomial has a root in the ball
v_L(z - c) ≥ r have measure q ^ (-(r + d)), where q is the residue cardinality of K. Then
the root count of L integrates over the Eisenstein region to q ^ (-(d + 1)) (1 - q⁻¹).
The integral of the root count (Serre's Lemmas 2 and 3). Let L ∈ σ_K(n) have
discriminant exponent d, and let μ be a left-invariant measure on the coefficient space
Fin n → K, normalized so that the integer box has volume 1. Then the root count of L
integrates over the Eisenstein region to q ^ (-(d + 1)) (1 - q⁻¹), where q is the residue
cardinality of K.