Multivariate counting of wild abelian extensions

Fabian Gundlach

Multivariate counting of wild abelian extensions

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that for any finite abelian p-group G and any global function field K of characteristic p, the multivariate generating function counting sub-G-extensions by all successive higher-ramification height functions is rational…

desk verdict Solid, significant extension of Gun26 to arbitrary base curves and multivariate ramification heights; the main theorem holds and the only real caveat is an imported local lemma that deserves an explicit generality statement. read the letter →

arxiv 2607.27364 v1 pith:QHFG7NAE submitted 2026-07-29 math.NT

Fabian Gundlach This is my paper · ORCID

classification math.NT MSC 11R4511R3711S4030B1011S15

keywords wildabelianextensionsfunctionfieldshigherramificationgroupsrationalgeneratingfunctionsArtin-SchreierconductorHasse-Weilzetap-Selmergroupp-groups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Fix a finite abelian p-group G of exponent $p^e$ and a global function field $K$ of characteristic $p$. The paper establishes that the multivariate generating function counting sub-$G$-extensions $\varphi$ of $K$ by the $e$ degrees $\deg(\mathrm{exjump}_i(\varphi)-p\,\mathrm{exjump}_{i+1}(\varphi))$ is rational, and writes it explicitly as a finite sum of products of translates of the Hasse-Weil zeta function of the base curve. This turns a counting problem with infinitely many local contributions at every place into a closed-form object. The result gives exact, not merely asymptotic, formulas for the number of extensions with prescribed successive ramification heights for every base curve, together with a product asymptotic when all heights grow.

What carries the argument

The load-bearing object is the explicit filtration of the local pro-$p$ unit group $U_P^1 = 1+\pi_P O_P$. For $k=(k_0,k_1,\ldots)$ in the cone $J_e=\{k_i\ge pk_{i+1},\ k_i=0\text{ for }i\ge e\}$, define $\lambda_i(k)=\min\{\lambda\ge 0 : i>k_\lambda\}$ and $R_P^k=\prod_{i\ge 1,\,p\nmid i} p^{\lambda_i(k)} \mathbb{Z}_p^{d}$ with $d=\deg(P)$. Lemma 4.4 supplies an isomorphism $\alpha:R_P^0\xrightarrow{\sim} U_P^1$ that sends $R_P^k$ onto the subgroup generated by $(U_P^{k_j+1})^{p^j}$ for all $j$, converting the ramification-jump conditions $\mathrm{exjump}_{P,j}(\varphi)\le k_j$ into the vanishing of $\varphi$ on $R_P^k$. Lemma 4.10 then gives the local count $|\mathrm{Hom}(R_P^0/R_P^k,G)| = Q_P^{\sum (c_i v_i + d_{i+1} w_i)}$ when $k_i-pk_{i+1}=v_i+pw_i$. The $p$-Selmer group of $K$ controls the boundary map $\delta$ to $\mathrm{Ext}^1_{\mathbb{Z}}(\mathrm{Pic}^0_K,G)$, ensuring that the character-sum over the local obstructions differs from its naive Euler product only by finitely many places, which explains the finite correction polynomials $B_\chi$.

What would settle it

Take $K=\mathbb{F}_q(t)$, $G=C_p$, and compare the first coefficients of $Z_K(qX)Z_K(q^{p-1}X^p)/(Z_K(q^p X^p)Z_K(X))$ with a direct enumeration of Artin-Schreier extensions $y^p-y=f$ over $\mathbb{F}_q(t)$ ordered by their conductor; any disagreement in the coefficient of $X^n$ for a small $n$ would show that Theorem 6.3 computes the wrong generating function.

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Extended reading notes

Core claim

The central claim is Theorem 6.3: for every finite abelian group $G$ of exponent $p^e$ and every global field $K$ of characteristic $p$, $$F_K(X_0,\ldots,X_{e-1}) = \frac{1}{|G|} \sum_{\varphi\in\mathrm{Hom}(\Gamma_K,G)} \prod_{i=0}^{e-1} $X_i^{{\deg(\mathrm{exjump}}$_i(\varphi)-p\,\mathrm{exjump}_{i+1}(\varphi))}$$ is rational. More precisely, $F_K$ equals $\prod_{i=0}^{e-1} H_i(X_i)$ plus a finite sum over the nontrivial characters $\chi$ of $\mathrm{Ext}^1_{\mathbb{Z}}(\mathrm{Pic}^0_K,G)$ of $\big(\prod_{i=0}^{r_\chi-1} H_i(X_i)\big)\,\frac{1}{Z_K(q^{d_{r_\chi}}X_{r_\chi})}\,B_\chi(X_{r_\chi},\ldots,X_{e-1})$, where $H_i(X_i)=\frac{Z_K(q^{c_i}X_i)Z_K(q^{d_{i+1}}X_i^p)}{Z_K(q^{pc_i}X_i^p)Z_K(q^{d_i}X_i)}$ and $B_\chi$ is a polynomial with integer coefficients and constant term 1, which is identically 1 when the genus of $K$ is at most 1. Thus rationality survives for non-rational base fields and the count tracks all $e$ successive drops in the higher-ramification filtration at once.

Load-bearing premise

The proof rests on the imported lemma that the subgroup generated by $(U_P^{k_j+1})^{p^j}$ inside the local pro-$p$ unit group at $P$ is exactly the product $R_P^k$; if this filtration identification fails for some place type, the explicit rational formula would not be the true generating function.

Editorial extensions

If this is right

For any prescribed tuple $(n_0,\ldots,n_{e-1})$, the exact number of sub-$G$-extensions with heights $n_i$ is the coefficient of $\prod X_i^{n_i}$ in a known finite sum of Hasse-Weil factors, so counting is a finite algebraic operation once the curve's zeta function is known.
When all $n_i$ tend to infinity the counted number is asymptotic to $(\prod C_i) \prod q^{(1+c_i)n_i}$ and is bounded by a constant times the same product for every tuple, identifying the dominant pole $X_i=q^{-1-c_i}$ as the source of all exponential growth.
The single-variable Artin-Schreier-conductor generating function $F_K^{\mathrm{asc}}(X)=F_K(X,X^p,\ldots,X^{p^{e-1}})$ is rational and has a unique innermost simple pole at $X=q^{-a}$ with $a=1+\dim_{\mathbb{F}_p}G[p]$, giving the asymptotic $Cq^{an}$ for extensions counted by conductor degree.
For a base curve of genus at most one, all correction polynomials $B_\chi$ are equal to 1, so the rational formula reduces to the product of zeta-function translates with no additional finite factor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Substituting $X_i=Y^{a_i}$ for any nonnegative weights would give exact rational generating functions for any linear combination of the $e$ height functions, so the same closed-form phenomenon should hold for a whole family of coarser inertial invariants; the paper only carries out the substitution $X_i=X^{p^i}$ for the Artin-Schreier conductor.
A natural stress test is the non-abelian analogue: fix a non-abelian $p$-group $N$ and count surjective maps $\Gamma_K\to N$ by the same height functions. The paper's structure suggests the obstruction is whether a filtration like $R_P^k$ exists for the relevant pro-$p$ quotients, and if it does, rationality of the corresponding generating function would be a plausible extension.
Because the formula is built from the Hasse-Weil zeta function, the Riemann hypothesis for function fields implies that the corrections to the main asymptotic are oscillatory and controlled by the zeros of $Z_K$; this analytic refinement is not spelled out in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Fabian Gundlach proves that the multivariate generating function counting sub-G-extensions of an arbitrary global function field K (any genus, any constant field) by e ramification heights is rational, and gives an explicit formula in terms of the Hasse-Weil zeta function. This is a genuine upgrade over [Gun26], which covered only rational K and a single height. The theorem is new and, as far as I can tell, correct.

The proof is careful and mostly self-contained. The use of p-Selmer groups to control the boundary map into Ext^1(Pic^0_K,G) is well executed, and the correction of Lagemann's mistake (Remark 5.7) is a real service. The local recursion in Theorem 6.2 is intricate but the final closed form is credible; I checked small cases like G=C_p, e=1 and the elliptic curve example, and they come out right. The asymptotics in Theorem 7.3 and the single-variable corollary are clean and useful.

The one soft spot is the dependence on Lemma 4.4, imported from [Gun26, Lemma 4.1]. The paper proves the combinatorial identity (4.1) that makes the filtration explicit, but the existence of the isomorphism α with the right subgroups is cited, not reproved. If that lemma were only proved for rational K, or failed for residue degree d>1, the local counts and hence Theorem 6.3 would fall. I think the stress-test note overstates the risk: the lemma is published and local, and there's no indication it is restricted to the rational case. But I would ask the author to state explicitly that it holds for every place of every global function field, or give the one-paragraph proof. This is a clarification, not a fix.

The only other minor issue is readability: the recursion in Theorem 6.2 is dense, and the 'small trick' in equation (6.3) deserves a sentence or two more. A serious referee will follow it, but it pushed me.

Bottom line: this is a solid, significant paper. The main theorem is a real result with explicit content, not just an abstract rationality claim. It deserves a careful referee and publication. I'd recommend accepting it into the review process; if I were refereeing, I'd ask for the clarification above and otherwise recommend acceptance.

Referee Report

0 major / 5 minor

Summary. The paper studies the multivariate counting of wild abelian extensions of a global function field K of characteristic p. For a finite abelian p-group G of exponent p^e, it defines e height functions deg(exjump_i(φ) - p·exjump_{i+1}(φ)) and forms the generating function F_K(X_0,...,X_{e-1}) = (1/|G|) ∑_{φ∈Hom(Γ_K,G)} ∏_i X_i^{deg(exjump_i(φ)-p·exjump_{i+1}(φ))}. The main result, Theorem 1.1 / Theorem 6.3, asserts that this multivariate generating function is rational and gives an explicit formula as the product of the trivial-character Euler factor ∏_i H_i(X_i) plus a finite sum over nontrivial characters of Ext^1_Z(Pic^0_K,G), each term involving a partial product of the H_i and a reciprocal Hasse-Weil zeta factor times a polynomial B_χ. The proof proceeds by class field theory, reducing global counts to a boundary map δ into Ext^1_Z(Pic^0_K,G), then to local unit-group filtrations. The local counting engine combines a filtration of U^1_P (Lemma 4.4, building on [Gun26, Lemma 4.1]), a count of homomorphisms R^0_P/R^k_P→G (Lemma 4.10), and a Selmer-group comparison of the filtrations H^k_P and I_r (Section 5). The paper also derives single-variable rationality and pole/asymptotic statements for the Artin-Schreier conductor in Theorem 7.5.

Significance. If correct, this is a substantial advance: it gives exact, not merely asymptotic, multivariate counting formulas valid for every global function field, removing the rationality restriction of earlier work such as [Gun26] and introducing a genuinely multivariate height statistic. The explicit form in Theorem 6.3, with the Hasse-Weil zeta function and finitely many correction polynomials B_χ, is concrete enough to produce exact formulas for fixed K and G (see Example 6.4) and to yield uniform asymptotics and a negative-residue pole statement in Theorem 7.5. The paper also identifies and corrects a mistake in [Lag15] (Remark 5.7). The proof is detailed and internally coherent: the local generating function recursion in Theorem 6.2 is intricate but the key steps, including the Selmer obstruction and the boundary-map comparison, fit together. I checked the formulas in the small case G=C_p with e=1 against the elliptic-curve example and the local count of Lemma 4.10, and they agree.

minor comments (5)

[Definition 4.2 and Definition 4.7] The letter d is used both for the residue degree of a place in Definition 4.2 and for the sequence d_i in Definition 4.7. Consider using d_P or r_P for the residue degree to avoid a notational collision that could confuse readers of Lemmas 4.4 and 4.10.
[Lemma 4.4 proof] The existence of the isomorphism α is imported from [Gun26, Lemma 4.1], and the paper proves only the coordinate identity (4.1). Since all of the subsequent local counting depends on this lemma, please state explicitly in the text that [Gun26, Lemma 4.1] applies to completions of an arbitrary global function field of characteristic p, including all residue degrees d≥1; if that lemma was originally stated only for d=1, a proof for general d should be supplied.
[Theorem 6.2, Equation (6.3)] The tuple notation "(k_0+p^g,...,k_{g-1}+p,k_g,k_{g+1},...)" is not well-defined for g=0. The accompanying sentence indicates that the g=0 map is the identity, but the display should be adjusted, for example by writing the map for g≥1 and treating g=0 separately, to prevent a misreading of the induction.
[Corollary 5.5] The condition "k_0≥...≥k_{r-1}>0 = k_r = k_{r+1}=..." is terse and could be misread. Rephrase as "k_0≥...≥k_{r-1}≥1 and k_j=0 for all j≥r" for clarity.
[Lemma 7.1(a)] The positivity of C_i is asserted to follow from Remark 4.9(b)(c) together with the properties of Z_K. Please state explicitly that Z_K has no zeros in the closed disc |X|≤q^{-1}, since this standard fact is used to justify the nonzero denominators and the sign computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, and the only imported prior-work lemma is an independent published structural statement about local unit groups.

full rationale

The paper derives the multivariate generating function F_K from first principles. The chain starts with class field theory (Section 3, exact sequence (3.2)), the p-Selmer group estimates (Section 2), and an explicit local count of homomorphisms (Lemma 4.10). The local count in turn rests on Lemma 4.4, whose proof combines a coordinate computation (Equation (4.1), proved in the paper) with the isomorphism and filtration statement imported from [Gun26, Lemma 4.1]. That citation is to the author's prior published work, but it is a parameter-free structural statement about pro-p unit groups of local function fields and does not assume or depend on the present theorem; it is therefore independent support under the review rules, not a circular reduction. No parameter is fitted to data and then renamed as a prediction; the exponents c_i, d_i in the Euler factors are computed from the group structure of G (Definition 4.7) and the residue field size, not fitted. No uniqueness theorem from the author's prior work is invoked to forbid alternatives, and no known result is merely renamed. The final rationality statement is assembled from Euler products of rational Hasse-Weil zeta factors by finite character sums, and the pole/asymptotic analysis in Section 7 proceeds forward from these formulas. Even the single-variable specialization F_K(X,X^p,...,X^{p^{e-1}}) is derived by the identity asc(phi)=sum (exjump_i - p exjump_{i+1}) p^i, not by definition of the multivariate function. The only caveats are external-dependency concerns: if [Gun26, Lemma 4.1] failed in some place type, Theorem 6.3 would fail; but that is a question of correctness of a citation, not circularity of the derivation. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; all constants c_i,d_i are functions of the group G. The proof uses standard class field theory, the Hasse-Arf theorem, the Hasse-Weil zeta function, and a cited filtration lemma for local unit groups. No new physical or mathematical entities are postulated.

assumptions (5)

standard math Local and global class field theory for global function fields, including the idele class group exact sequence used in Section 3.
Used to identify Hom(Gamma_K,G) with maps on A_K^times/K^times and to compute exjump_i through unit groups.
standard math The Hasse-Arf theorem: jumps in the higher ramification filtration of abelian extensions occur right after integers.
Justifies the integrality and the equality between an infimum over real v and a minimum over integers in the definition of exjump_P,i.
standard math The unit-group filtration lemma [Gun26, Lemma 4.1] giving explicit coordinates of U_P^1 as a product of copies of Z_p.
This is the base for Lemma 4.4; all local counts and the exponents c_i,d_i flow from this cited filtration description.
standard math The Hasse-Weil zeta function of a global function field is rational and has the standard pole and zero locations, including the Riemann hypothesis for function fields.
Used in Section 7 to locate the innermost pole of H_i at X_i=q^{-1-c_i} and to control the remaining factors in the convergence arguments.
standard math Lemma 2.1, taken from [Pot26, Proposition 5.2b], describing representatives in O_P^times/O_P^{times p}.
Used in Lemma 2.3 and Remark 2.2 to compute omega_P([y]) and the divisor of the logarithmic derivative.

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Cite this review

Pith. "Pith review of Multivariate counting of wild abelian extensions." pith.science (2026). https://pith.science/paper/QHFG7NAE

@misc{pith2026260727364,
  author       = {Pith},
  title        = {Pith review of: Multivariate counting of wild abelian extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QHFG7NAE}},
  note         = {Machine review of arXiv:2607.27364}
}

abstract

Let $K$ be a global function field of characteristic~$p$ and let $G$ be a finite abelian group of exponent $p^e$. We show that the multivariate generating function counting sub-$G$-extensions of $K$ with respect to $e$ specific height functions (encoding successive drops in the exponents of the higher ramification groups) is rational.

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Reference graph

Works this paper leans on

15 extracted references · 4 canonical work pages

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Ellenberg and Akshay Venkatesh

Jordan S. Ellenberg and Akshay Venkatesh. Counting extensions of function fields with bounded discriminant and specified G alois group. In Geometric methods in algebra and number theory , volume 235 of Progr. Math. , pages 151--168. Birkh\" a user Boston, Boston, MA, 2005. https://doi.org/10.1007/0-8176-4417-2_7 doi:10.1007/0-8176-4417-2_7

work page doi:10.1007/0-8176-4417-2_7 2005

[2] [2]

Analytic combinatorics

Philippe Flajolet and Robert Sedgewick. Analytic combinatorics . Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511801655 doi:10.1017/CBO9780511801655

doi:10.1017/cbo9780511801655 2009

[3] [3]

Counting two-step nilpotent wildly ramified extensions of function fields, 2025

Fabian Gundlach and Béranger Seguin. Counting two-step nilpotent wildly ramified extensions of function fields, 2025. https://arxiv.org/abs/2502.18207 arXiv:2502.18207

arXiv 2025

[4] [4]

Lifts of unramified twists and local-global principles, 2026

Fabian Gundlach and Béranger Seguin. Lifts of unramified twists and local-global principles, 2026. https://arxiv.org/abs/2603.15544 arXiv:2603.15544

work page arXiv 2026

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Malle's conjecture with multiple invariants, 2022

Fabian Gundlach. Malle's conjecture with multiple invariants, 2022. https://arxiv.org/abs/2211.16698 arXiv:2211.16698

arXiv 2022

[6] [6]

Counting abelian extensions by A rtin- S chreier conductor

Fabian Gundlach. Counting abelian extensions by A rtin- S chreier conductor. Proc. Amer. Math. Soc. , 154(2):527--540, 2026. https://doi.org/10.1090/proc/17440 doi:10.1090/proc/17440

work page doi:10.1090/proc/17440 2026

[7] [7]

Equidistribution for abelian extensions of global fields

Jiazhi He. Equidistribution for abelian extensions of global fields, 2026. https://arxiv.org/abs/2607.16079 arXiv:2607.16079

work page Pith review arXiv 2026

[8] [8]

Distribution of A rtin- S chreier extensions

Thorsten Lagemann. Distribution of A rtin- S chreier extensions. J. Number Theory , 132(9):1867--1887, 2012. https://doi.org/10.1016/j.jnt.2012.03.011 doi:10.1016/j.jnt.2012.03.011

doi:10.1016/j.jnt.2012.03.011 2012