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Counting abelian extensions of global function fields, wild parts included, reduces to an explicit asymptotic formula with a computable leading constant and a power of log M.

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2026-08-01 21:25 UTC pith:V2UGXMSV

load-bearing objection Strong paper: the main theorem is credible and the pole analysis matches (1.4); the real problem is that Theorem 2.2 is cited to a MathOverflow answer, not proved. the 2 major comments →

arxiv 2607.16079 v2 pith:V2UGXMSV submitted 2026-07-17 math.NT

Equidistribution for abelian extensions of global fields

Jiazhi He This is my paper
classification math.NT MSC 11R11R4211R4511M41
keywords abelian extensionsfunction fieldsMalle's conjectureequidistributionfrobenian functionsArtin L-functionsconductorlocal conditions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes an asymptotic formula for the number of abelian extensions of a global function field with a given finite abelian Galois group, counted by conductor and subject to prescribed local conditions. The formula is uniform in the group, including groups with a non-trivial wild p-part, and gives an explicit leading constant and an explicit exponent of log M. This completes the abelian case of Malle's conjecture over function fields with local conditions, and the proof introduces a theory of frobenian functions adapted to constant field extensions. The result is interpreted as equidistribution on the classifying stack BG, agreeing with a Tamagawa measure prediction for tame groups.

Core claim

Theorem 1.3: for a nontrivial finite abelian group G and a global function field k, the averaged counting function Nav(k,G,W,q^M) is asymptotic to q^{a(G)M} c(k,G,W) M^{b(k,G)-1} as M→∞, where a(G) and b(k,G) are explicit invariants of k and G and c(k,G,W) is a product of local factors, a residue of an Artin L-function, and a rational number depending on the wild part. The average over a short block of exponents is necessary because the generating series has multiple poles of maximal order; the averaging cancels their phases. If the paper is right, it gives the full abelian case of Malle's conjecture over function fields with local conditions, including the case where G has a non-trivial wil

What carries the argument

The generating series FG,W(s) is analysed through a harmonic-analytic decomposition into local Fourier transforms; the global transform is evaluated by Poisson summation over S-units. A new theory of frobenian functions over function fields, together with a theorem on Artin L-functions (holomorphic for non-constant characters, with an explicit formula for constant characters), expresses the transform as a product of Dedekind zeta functions of constant extensions. The rightmost poles of this product, and their orders, are computed explicitly; a Tauberian theorem then converts the pole data into the asymptotic formula. The averaging in Nav removes the phase factors of multiple maximal poles so

Load-bearing premise

The whole pole computation rests on the theorem that non-constant Artin L-functions over function fields are holomorphic; the paper does not prove this and cites it to an external online source.

What would settle it

Compute the generating series for a concrete wild example, say k=F_q(T) and G=C_p×C_p, for a modest number of terms and compare the averaged counts with the predicted asymptotic; alternatively, find a non-constant character of a geometric extension of a function field where L(χ,s) has a zero or pole at Re(s)=1/2, which would contradict the cited holomorphy theorem.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Counts abelian extensions of any global function field with any finite abelian Galois group, including wild groups, with prescribed local conditions.
  • Agrees with the stack-theoretic Tamagawa measure prediction for tame groups, and provides new explicit constants for wild groups.
  • Implies a local-global principle: any finite set of local abelian extensions can be glued to a global one, contrasting with the number field case.
  • Establishes equidistribution of local behaviours for abelian extensions over function fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The frobenian-function framework developed here likely applies to other counting problems over function fields, such as counts with respect to different heights or for non-abelian groups once suitable holomorphy statements are available.
  • The explicit rational factor appearing for wild groups may point to a general formula for the effective-cone constant in Malle's conjecture over function fields.
  • The dependence of the leading constant on the constant field through [k(μ_α):k] could be tested numerically for small q and small groups to verify the predicted dependence.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a harmonic-analytic and frobenian framework over global function fields and uses it to prove asymptotic estimates for the averaged counting function N_av(k,G,W,q^M) of abelian G-extensions ordered by conductor, with finitely many prescribed local conditions. The main results are an equidistribution statement (Theorem 1.1), an explicit asymptotic with leading constant (Theorem 1.3), and a stack-theoretic interpretation via Tamagawa measures on BG (Theorem 1.4). The proof proceeds through the conductor Dirichlet series, local Fourier transforms, Poisson summation over ideles, and a pole analysis in which the constant subextension contributes extra zeta factors.

Significance. If the main theorem is established, it gives the full abelian case of Malle's conjecture over global function fields with local conditions and explicit leading constants, including wild Galois groups, and connects the count with an equidistribution statement on algebraic stacks. The local convergence analysis of §3 and the explicit treatment of wild ramification are substantial technical contributions, and the averaged counting device is a natural way to handle the multiple rightmost poles in the function-field setting. The paper is therefore potentially important for the analytic theory of abelian extensions over function fields and for ongoing work on Malle's conjecture.

major comments (2)
  1. [§2.1, Theorem 2.2 and Proposition 2.9] Theorem 2.2 is load-bearing: Proposition 2.9 uses it to pass from the class-function decomposition to the factorization F(s)=ζ_M(s)^{m(ρ)}P(s) in (2.10)-(2.13), and all pole orders and the leading constant in Theorem 1.3 depend on that factorization. The proof given in the manuscript only establishes the constant-character formula (2.4); the holomorphy and Riemann-hypothesis assertion for non-constant characters is deferred to [Saw26], a MathOverflow answer, with the remark that [MS94, Lemma 1] has a gap. This is not an acceptable basis for a central theorem in a journal paper. Please include a complete proof of Theorem 2.2 in the paper or replace [Saw26] with a peer-reviewed reference that contains the full argument.
  2. [§4.3, Lemma 4.4 and surrounding argument] The proof of the analytic continuation of the global Fourier transforms, especially the reduction of non-trivial x to the frobenian function ρ_x of (4.6), uses Lemma 4.4, whose proof for t=0 is cited to [Alf25, Coro. 6.15], a preprint. Since this lemma controls the decomposition in Lemma 4.6 and hence Proposition 4.7, the dependence should be clarified. If [Alf25] is not yet published, include a short proof of the needed statement or state it as an explicit lemma in the present paper.
minor comments (4)
  1. [§2.2, proof of Lemma 2.6] In the displayed fact after (2.6), the conclusion should be phrased as 'z_i is a non-negative real number' rather than 'z_i=|z_i|', since z_i is complex. The intended content is clear.
  2. [§3.3, Lemma 3.5 and Proposition 3.7] The notation v_p(j-1) is used for the p-adic valuation, but v is also used for a place. This is typographically confusing; please use ord_p or another symbol.
  3. [§1, equation (1.3) and Corollary 4.10] The reduction of the gcd over all α| |G_t| to the gcd over prime ℓ in (1.3) is stated tersely in Corollary 4.10. A one-sentence justification that [k(μ_ℓ):k] divides [k(μ_α):k] for each prime divisor ℓ of α would improve readability.
  4. [§6.1] Typo: 'immediatte' should be 'immediate'.

Circularity Check

0 steps flagged

No significant circularity: all constants (a(G), b(k,G), c) are derived from local Fourier analysis and zeta pole data, not fitted; the only external reliance (Theorem 2.2) is a correctness risk, not a self-citation or definitional circularity.

full rationale

The paper's derivation of Theorem 1.3 is not circular. The weight a(G) in the averaged count (1.2) is not an input to the asymptotic: Proposition 3.7 and Lemma 3.10 derive a(G) as z_{p^e}(G), the maximal pole/abscissa location of the local Fourier transforms, and Lemma 4.9/Corollary 4.10 independently compute the pole orders b(k,G) and the set B(k,G). The leading constant c(k,G,W) is obtained by evaluating the holomorphic function D at s=a(G) via Euler products (Proposition 4.3, §5), rather than by fitting. The quantities d(k,α), M_G, and λ_v are defined from cyclotomic constant extensions and Artin L-functions, independent of the counting function. The only load-bearing external input is Theorem 2.2 (non-constant Artin holomorphy), whose proof is deferred to the MathOverflow answer [Saw26]; this is a correctness risk, not a circularity, since [Saw26] is not by the present author and the asserted holomorphy is not equivalent to the counting asymptotics by construction. There are no self-citations by the author, so the self-citation patterns do not apply.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

The central derivation assumes class field theory, the Weil/Rosen zeta-function facts, Chebotarev, and a Tauberian theorem; these are standard. The non-standard load-bearing inputs are Theorem 2.2 (complete proof in a MathOverflow answer, not in the paper), Lemma 4.4 (an arXiv preprint), and Lagemann's Lemma 4.8. No numbers are fitted; a(G), b(k,G) and d(k,α) are functions of G and k. The averaging weight in N_av is a bookkeeping device, not a free parameter.

axioms (9)
  • standard math Global class field theory: Hom(Γ_k,G) ≅ Hom(A_k^*/k^*,G) and local Artin reciprocity Hom(Γ_{k_v},G) ≅ Hom(k_v^*,G).
    Used in §3.1 to rewrite the conductor Dirichlet series as a sum over idèle class characters; the counting relies on this identification.
  • standard math Zeta function properties of global function fields: ζ_k(s)=L(q^{-s})/((1-q^{1-s})(1-q^{-s})) and the Riemann hypothesis |roots|=q^{-1/2} (Lemma 2.1, from Rosen).
    Used throughout §2–4 to locate poles of Euler products and to bound zeros.
  • domain assumption Theorem 2.2 (Artin holomorphy for function fields): L(χ,s) is holomorphic (polynomial in u) iff χ is non-constant; constant χ gives Z(χ,u)=Z_k(χ(Frob_q)u).
    The paper sketches the constant-character case but outsources the non-constant holomorphy proof to [Saw26], a MathOverflow answer; this factorization is essential to Prop 2.9.
  • standard math Chebotarev density theorem (used to conclude pointwise identity of class functions from identity on Frobenius classes).
    Invoked in Lemma 2.6(2)(c)⇒(b).
  • standard math Tauberian theorem for power series with poles (Odlyzko, Thm 11.2).
    Section 5 converts the pole analysis of F_G,W(s) into the asymptotic for N and N_av.
  • standard math Grunwald–Wang / Hasse norm principle for abelian extensions over function fields ([NSW08, Thm 9.1.11]).
    Used in Prop 4.7(1) to ensure x_t,v ∈ k_v^{*|G_t|} for almost all v implies x_t=1_t.
  • domain assumption Lemma 4.4: if x∈k^{*p^t}∖k^{*p^{t+1}}, its v-adic expansion has non-zero a_{p^t,v} for all but finitely many v (cited to [Alf25, Coro. 6.15]).
    Load-bearing for the shape of local Fourier transforms for non-trivial x; proof not included.
  • domain assumption Lagemann's Lemma 4.8 (ε(j) positivity) as stated in [Lag15, Prop. 6.4].
    Used to identify the rightmost poles with p^f|j; the paper notes it is equivalent to Lagemann's rank formulation but does not reprove it.
  • domain assumption Cesnavicius [Ces15, Prop. 3.5]: H^1(k_v,G) is discrete.
    Used in the proof of Theorem 1.4 to decompose any continuity set into basic clopen sets.

pith-pipeline@v1.3.0-alltime-deepseek · 30598 in / 33203 out tokens · 243538 ms · 2026-08-01T21:25:49.332800+00:00 · methodology

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read the original abstract

We establish asymptotic formulas for abelian extensions of global function fields ordered by conductor and subject to prescribed local conditions. Our proof combines harmonic analysis with a theory of frobenian functions over global function fields developed in this paper. We interpret our result via equidistribution on algebraic stacks.

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