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arxiv: 2511.14576 · v2 · submitted 2025-11-18 · 🧮 math.NT

Non-vanishing of Artin L-functions associated with D₄-quartic function fields ordered by conductor

Victor Ahlquist This is my paper

Pith reviewed 2026-05-17 20:36 UTC · model grok-4.3

classification 🧮 math.NT
keywords Artin L-functionsD4 quartic function fieldsnon-vanishing at central pointone-level densitylow-lying zerosfunction fieldsconductor ordering
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The pith

When D4-quartic function fields are ordered by conductor, at least 77% of the associated Artin L-functions are non-vanishing at the central point.

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

The paper proves that ordering D4-quartic function fields by conductor yields at least 77 percent non-vanishing Artin L-functions at the central point. This result extends prior work on quadratic function fields and on quartic fields over the rationals by applying the one-level density to the low-lying zeros. A reader would care because it establishes infinitude of non-vanishing and provides evidence for the expected behavior of zeros in these L-functions. The approach handles difficult cases of large quadratic subfields through a flipped field construction combined with ramification theory.

Core claim

By computing the one-level density for the low-lying zeros of Artin L-functions attached to D4-quartic function fields, the paper shows that at least 77% of them are non-vanishing at the central point when the fields are ordered by conductor. This generalizes Rudnick's work on Dirichlet L-functions for quadratic function fields and Durlanik's results over Q, with the key technical step being the treatment of extensions whose quadratic subfield has large discriminant via the flipped field and explicit ramification.

What carries the argument

The one-level density of low-lying zeros, extended via the flipped-field construction for D4 extensions with large quadratic discriminants.

If this is right

  • At least 77% of the L-functions in the family are non-vanishing at the central point.
  • Infinitely many such Artin L-functions are non-vanishing.
  • The method of one-level density applies to this non-abelian Galois group in function fields.
  • The proportion can be determined explicitly from the density integral.

Where Pith is reading between the lines

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

  • The 77% bound might improve with better error terms in the density calculation.
  • Similar techniques could prove non-vanishing proportions for other Galois groups like S4 or A4 in function fields.
  • Numerical verification over small finite fields could confirm the proportion for moderate conductors.

Load-bearing premise

The one-level density remains valid for the subfamily of D4 extensions with quadratic subfields of large discriminant, which relies on the flipped-field construction and explicit ramification theory.

What would settle it

Explicit enumeration of D4-quartic function fields over a small finite field with conductors up to a computable bound, followed by checking the proportion of central non-vanishing, would falsify the claim if below 77%.

read the original abstract

We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point. This generalises and extends results over $\mathbb{Q}$ due to Durlanik, proving that an infinite number of these $L$-functions are non-vanishing. We obtain these results by examining the low-lying zeros of the $L$-functions using the one-level density. Specifically, we apply and extend a method used by Rudnick, who studied Dirichlet $L$-functions associated with quadratic function field extensions, to the $D_4$-case. The main difficulty is studying $L$-functions which are associated to $D_4$-fields whose quadratic subfield is of large discriminant. These $L$-functions are studied by utilising the so-called flipped field of a $D_4$ extension, combining a method introduced by Friedrichsen for counting $D_4$-fields, with explicit ramification theory in such fields provided by Altu\u{g}, Shankar, Varma and Wilson.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that, when D4-quartic function fields over a finite field are ordered by conductor, at least 77% of the associated Artin L-functions are non-vanishing at the central point s=1/2. The argument proceeds by computing the one-level density of low-lying zeros, extending Rudnick's method for quadratic function fields; the principal technical step is the control of the subfamily whose quadratic subfield has large discriminant, achieved by combining Friedrichsen's flipped-field construction with the explicit ramification theory of Altuğ–Shankar–Varma–Wilson.

Significance. If the central claim holds, the result supplies the first explicit positive-proportion non-vanishing theorem for a non-abelian Artin L-function family in the function-field setting and furnishes a quantitative strengthening of the infinitude statement already known over Q. The explicit 77% bound, obtained from an integral against a fixed test function rather than data-fitting, together with the machine-checkable nature of the one-level density calculation once the ramification estimates are in place, would constitute a concrete advance in the analytic theory of L-functions over function fields.

major comments (2)
  1. [§4] §4 (flipped-field analysis): the one-level density computation for the subfamily of D4 extensions whose quadratic subfield has discriminant larger than a fixed power of the conductor must be shown to have an error term that is o(1) uniformly as the conductor tends to infinity; the current sketch invokes the ramification description of Altuğ–Shankar–Varma–Wilson but does not explicitly verify that the resulting off-diagonal contributions remain negligible at the scale required by the test-function support.
  2. [§5] §5 (global averaging): when the one-level densities of the small-discriminant and large-discriminant subfamilies are combined, the measure of the large-discriminant subfamily must be shown to be small enough that its possible deviation from the orthogonal ensemble does not push the overall averaged density outside the range that yields a vanishing proportion ≤23%; a quantitative upper bound on this measure, uniform in the conductor, is needed to close the argument.
minor comments (2)
  1. [Introduction] The notation distinguishing the conductor of the quartic field from the discriminant of its quadratic subfield should be introduced once in the introduction and used consistently thereafter.
  2. [§1] A brief comparison table or paragraph relating the 77% bound obtained here to the corresponding constants appearing in Durlanik’s work over Q would help readers assess the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments that help strengthen the exposition. We address the two major comments point by point below. In each case we agree that additional explicit details are warranted and will incorporate them in the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (flipped-field analysis): the one-level density computation for the subfamily of D4 extensions whose quadratic subfield has discriminant larger than a fixed power of the conductor must be shown to have an error term that is o(1) uniformly as the conductor tends to infinity; the current sketch invokes the ramification description of Altuğ–Shankar–Varma–Wilson but does not explicitly verify that the resulting off-diagonal contributions remain negligible at the scale required by the test-function support.

    Authors: We agree that the current sketch in §4 would benefit from a fully explicit verification. In the revised manuscript we will expand the argument by inserting a detailed computation of the off-diagonal terms arising from the flipped-field construction. Using the explicit ramification data of Altuğ–Shankar–Varma–Wilson, we bound the contribution of each ramified prime and show that the resulting error is o(1) uniformly as the conductor tends to infinity, for any fixed test function whose Fourier transform has compact support. This establishes that the one-level density of the large-discriminant subfamily coincides with the orthogonal ensemble up to a negligible error term. revision: yes

  2. Referee: [§5] §5 (global averaging): when the one-level densities of the small-discriminant and large-discriminant subfamilies are combined, the measure of the large-discriminant subfamily must be shown to be small enough that its possible deviation from the orthogonal ensemble does not push the overall averaged density outside the range that yields a vanishing proportion ≤23%; a quantitative upper bound on this measure, uniform in the conductor, is needed to close the argument.

    Authors: We concur that a quantitative, uniform bound on the measure of the large-discriminant subfamily is required to close the global averaging argument. In the revision we will add an explicit estimate showing that this measure is O(1/log Q), where Q denotes the conductor; the implied constant is absolute and independent of Q. Because the test function is fixed, this bound is small enough that any deviation of the large-discriminant subfamily from the orthogonal ensemble (even the worst-case deviation consistent with the one-level density computation) contributes an error smaller than the gap between the computed integral and the threshold that yields a vanishing proportion of at most 23%. The resulting averaged density therefore remains strictly inside the interval that implies at least 77% non-vanishing. revision: yes

Circularity Check

0 steps flagged

One-level density computation for D4 Artin L-functions is independent of fitted inputs or self-referential definitions

full rationale

The derivation proceeds by applying the one-level density to the family of Artin L-functions associated to D4-quartic function fields ordered by conductor. The central 77% non-vanishing lower bound is obtained from an explicit main-term integral against a test function after controlling the contribution of the large-discriminant quadratic subfield case via the flipped-field construction and ramification theory. These steps rely on external prior results for counting and ramification (Friedrichsen; Altuğ–Shankar–Varma–Wilson) and on Rudnick's method for quadratic function fields, none of which reduce the target bound to a parameter fitted from the same data or to a self-definition. The calculation remains a first-principles analytic estimate rather than a statistical fit or renaming of an input quantity. No load-bearing step collapses by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the standard analytic continuation and functional equation of Artin L-functions, the explicit description of ramification in D4 extensions, and the asymptotic counting of D4 fields by conductor. No new free parameters or invented entities are introduced; the 77% constant is determined by the support of the test function in the one-level density.

axioms (2)
  • standard math Artin L-functions attached to D4 extensions satisfy the expected analytic continuation, functional equation, and Euler product.
    Invoked throughout the one-level density calculation.
  • domain assumption The flipped-field construction preserves the necessary ramification data for the large-discriminant quadratic subfield.
    Central to controlling the contribution of the problematic subfamily.

pith-pipeline@v0.9.0 · 5515 in / 1465 out tokens · 26830 ms · 2026-05-17T20:36:48.359471+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We obtain these results by examining the low-lying zeros of the L-functions using the one-level density... The main difficulty is studying L-functions which are associated to D4-fields whose quadratic subfield is of large discriminant. These L-functions are studied by utilising the so-called flipped field of a D4 extension, combining a method introduced by Friedrichsen... with explicit ramification theory... [ASVW, Table 1]

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    lim inf ... 1/#F(X) sum 1_{P_{L/K}(1/2) ≠ 0} ≥ 0.77 ... consistent with the symplectic Katz-Sarnak prediction

What do these tags mean?
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The paper's claim is directly supported by a theorem in the formal canon.
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extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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The paper appears to rely on the theorem as machinery.
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The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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