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REVIEW 3 major objections 3 minor 33 references

Murmurations of quadratic and cubic characters over function fields

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For the quadratic-character family over polynomial rings, the murmuration density is exactly 1/(q-1), and the same constant is the arithmetic correction in the trace statistics.

desk verdict Theorem 1.2's error term diverges under the stated hypotheses, so the main trace asymptotic is not proved as written; the paper's two density theorems are solid and worth engaging. read the letter →

arxiv 2608.01337 v1 pith:FI32PJMS submitted 2026-08-02 math.NT

Matilde Lalín , Kyu-Hwan Lee , Thomas Oliver , Alexey Pozdnyakov This is my paper · ORCID
classification math.NT MSC 11M5011R5811G20
keywords murmurationdensityquadraticcharactersfunctionfieldsFrobeniusclasshomologicalstabilityone-levelcentralpointnonvanishingcubic
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

This paper studies the murmuration phenomenon for families of Dirichlet characters over the polynomial ring $\mathbb{F}_q[T]$. Its first result is that for every monic irreducible polynomial $P$ of degree $2g$, the average of $\chi_D(P)\sqrt{|P|}$ over monic squarefree $D$ of degree $2g+1$ is exactly $1/(q-1)$; the murmuration density is therefore the same constant at every prime in the family. The same constant appears as a correction to the random-matrix prediction for the average trace of the Frobenius class at the transition power $n=2g$, and the paper evaluates that correction cohomologically through the stable homology of the moduli space of genus-$g$ hyperelliptic curves with one marked Weierstrass point. A consequence is a one-level density with a lower-order $-\hat f(1)/(g(q-1))$ term and a 100% non-vanishing theorem for $L(1/2,\chi_D)$ in the double limit $q\to\infty$ after $g\to\infty$. For the thin Kummer family of primitive cubic characters, the density is instead asymptotically the root number $\omega(\chi_P)$, showing that non-self-dual families can have a genuinely $P$-dependent murmuration shape.

What carries the argument

The load-bearing object is the murmuration density $M_{q,g}(P)=\langle\chi_D(P)\sqrt{|P|}\rangle$ on primes $P$ of degree $2g$ in the hyperelliptic ensemble. The mechanism is an explicit formula linking traces of Frobenius powers to character sums, together with a decomposition of $\operatorname{Tr}(\Theta^n)$ into symplectic irreducible characters organized by hooks $(a,1^b)$. On the cohomological side, the averages pass by the Grothendieck trace formula to étale homology of $\mathcal{H}^{1,0}_g$, the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point; a uniform homological stability theorem separates a stable range, where a plethystic generating function evaluates the homology explicitly, from an unstable range controlled by dimension bounds. The trivial character supplies the random-matrix main term for the compact symplectic group $\mathrm{USp}(2g)$, while the hook $(a,1^a)$ family produces the constant correction $1/(q-1)$.

What would settle it

Fix a small odd prime power $q$ and compute $M_{q,g}(P)$ for several primes $P$ of degree $2g$ as $g$ grows; any value that differs from $1/(q-1)$ by more than a decaying finite-field error would refute Theorem 1.1, and so would any failure of $\langle\operatorname{Tr}\Theta_D^{2g}\rangle$ to approach $-1-1/(q-1)$.

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Extended reading notes

Core claim

The central discovery is that in the hyperelliptic ensemble the murmuration density is exactly constant: $M_{q,g}(P)=1/(q-1)$ for every $P\in\mathcal{P}_{2g}$. Equivalently, the expectation of this density is the arithmetic correction in the trace formula: Theorem 1.2 gives, as $g\to\infty$, $\langle\operatorname{Tr}\Theta_D^n\rangle=\int_{\mathrm{USp}_{2g}}\operatorname{Tr} U^n\,dU - \mathbf{1}_{n=2g}/(q-1) + O(\mathbf{1}_{2|n}2^{n/2}q^{-n/8} + g^2 6^{g+n}q^{-\lfloor 2Ag+A+B\rfloor/2})$, under $2\nmid q$ and $q>2\max\{4,3/A+2\}$ with $A,B$ from the uniform stable range. The proof expresses $\operatorname{Tr}(\Theta^n)$ as a signed sum over symplectic hook characters, uses the Grothendieck trace formula to pass to étale homology of the moduli space $\mathcal{H}^{1,0}_g$, and isolates the trivial character as the random-matrix main term; the correction term at $n=2g$ is exactly the hook contribution evaluated by the stable-homology generating function. The same computation yields the one-level density and the nonvanishing corollary.

Load-bearing premise

The central result assumes that the error estimates from the parts of the cohomology that have not yet stabilized are small enough to isolate the constant correction; as written those errors grow with $n$, so the advertised trace asymptotics hold only when $q$ is large relative to $n$, or when $n$ is fixed.

Editorial extensions

If this is right

  • For every prime $P$ of degree $2g$, the expectation over $D\in\mathcal{H}_{2g+1}$ of $\chi_D(P)\sqrt{|P|}$ equals $1/(q-1)$, so the quadratic murmuration density is prime-independent.
  • The average trace $\langle\operatorname{Tr}\Theta_D^n\rangle$ matches the random-matrix prediction for $\mathrm{USp}(2g)$ except for the exact correction $-1/(q-1)$ at $n=2g$; odd-power traces vanish, and powers $n>2g$ see only the error term.
  • The one-level density acquires the lower-order term $-\hat f(1)/(g(q-1))$, and for $q$ large enough the Fourier support of the test function can be taken arbitrarily wide.
  • In the double limit $q\to\infty$ after $g\to\infty$, all but a vanishing proportion of the quadratic $L$-functions $L(1/2,\chi_D)$ are nonzero.
  • For the Kummer family of primitive cubic characters with $d=2n$, the murmuration density tends to the root number $\omega(\chi_P)$, so the density is a nontrivial function of $P$.

Reading between the lines

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

  • An implication the authors leave implicit is that, within the quadratic function-field family, the murmuration signal is a flat background rather than a $P$-dependent curve: every degree-$2g$ prime carries the same density, and any shape seen in number-field analogues would have to come from logarithmic-scale effects that have no direct function-field counterpart.
  • Applying the same cohomological machinery to subfamilies of cubic characters with fixed root number would test whether the asymptotic factor $\omega(\chi_P)$ in the Kummer result splits cleanly from a constant density; the paper notes this subfamily idea but does not develop it.
  • A sharper bound on the unstable cohomology range, if it exists, would let the $n=2g$ transition be resolved for fixed $q$ rather than $q$ large relative to $n$; the persistent constant $1/(q-1)$ would then be an arithmetic feature, not a large-$q$ artifact.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies murmuration densities for quadratic and cubic characters over F_q[T]. Theorem 1.1 evaluates the quadratic murmuration density pointwise as 1/(q-1). Theorem 1.2 claims large-genus asymptotics for the average traces Tr(Theta_D^n), displaying the random matrix theory main term together with the correction -1_{n=2g}/(q-1), and derives this from stable homology of the moduli space H_g^{1,0} via the results of BDPW23 and MPPRW24. Corollary 1.3 derives the one-level density with a lower-order correction, and Corollary 1.4 derives a full-density nonvanishing statement at the central point in the double limit q -> infinity after g -> infinity. Theorem 1.5 computes the Kummer cubic murmuration density using the published DFL22 Gauss sum asymptotics. The main novelty is the connection between the pointwise murmuration density, the Rudnick correction in the trace formula, and stable homology.

Significance. If Theorem 1.2 held as stated, it would be a substantial strengthening of Rudnick's trace asymptotics, with an explicit cohomological interpretation and applications to one-level density and nonvanishing. The proof of Theorem 1.1 is clean and elementary, and Theorem 1.5 is a straightforward consequence of DFL22. The paper uses no fitted parameters, and the reliance on DFL22 and MPPRW24 is not circular because those are independent inputs. However, the central trace asymptotic is not established under the stated hypotheses: the displayed error term fails to decay, and this failure propagates to Corollaries 1.3 and 1.4. The overall approach is sound in conception, but the main theorem needs a materially stronger quantification of q.

major comments (3)
  1. [Section 3.5 (Theorem 1.2)] The error term g^2 6^{g+n} q^{-theta(g)/2} does not tend to zero under the stated hypothesis q > 208. With A=1/34 and B=-35/34 one has theta(g)=floor(g/17 - 1), so the term is roughly g^2 6^n (6 q^{-1/34})^g q^{1/2}. For q=211, which satisfies q > 2 max{4,3/A+2}=208, the base 6*211^{-1/34} is about 5.12, which is larger than 1; the error grows exponentially even for fixed n=2. An O-term that tends to infinity cannot serve as a remainder in an asymptotic expansion, so the claimed main term, in particular the correction -1_{n=2g}/(q-1), is not isolated. The remark that the constants were not optimized does not repair the issue; a sufficient condition would be roughly q > 6^{34} for fixed n and q > 216^{34} for the n=2g transition. This is load-bearing, since Theorem 1.2 is the basis for the subsequent corollaries.
  2. [Section 3.3 (Proposition 3.19)] The advertised divergence originates in the unstable-range estimate. The proof bounds the tail by (2g)^2 6^{g+n} q^{-theta(g)/2} after using sum_{k>theta(g)} binom(2g,k) q^{-k/2} <= 2^{2g} q^{-theta(g)/2}. The estimate is algebraically valid, but under only q > 16 the factor 6 q^{-1/34} is larger than 1 (for q=17 it is about 5.5), so the displayed bound diverges as g grows. Consequently Proposition 3.19 as stated is not supported. The same divergent quantity appears in the error bounds of Propositions 3.20 and 3.21, so the proof of Theorem 1.2 cannot succeed without a much stronger, effectively n-dependent lower bound on q.
  3. [Section 4 (Corollaries 1.3 and 1.4)] Corollary 1.3 inherits the same defect: its error term contains v g^2 (6+2v)^g q^{-theta(g)/2}, which decays only when q > (6+2v)^{34}. The stated assumptions do not imply this for a fixed v, so the asymptotic expansion for the one-level density is not established as quantified. Corollary 1.4 then uses Corollary 1.3 to obtain lim inf_{g->infty} p_0(g) >= 1 - 1/(4v^2); with a divergent error no lower bound follows. The double-limit statement may be salvageable by taking q sufficiently large relative to v before letting v grow, but that is not what is stated.
minor comments (3)
  1. [Section 1.2 (Theorem 1.2)] The indicator notation '1 2|n' and '1 n=2g' should be typeset as 1_{2|n} and 1_{n=2g}; as printed they are difficult to read.
  2. [Sections 1.2 and 3.5] The lower bounds on q are stated inconsistently: Theorem 1.2 says q > 2 max{4,3/A+2}, while Proposition 3.20 appears to require q > 2^{3/A+2} if the superscript is intended. The authors should state a single precise quantification and verify it in every intermediate proposition.
  3. [Section 5] In the proof of Theorem 1.5, the factor (1+1/|P|)^{-1} appears in the main term in equations (5.2) and (5.3) but is omitted from the final statement; this difference is O(|P|^{-1}) and is absorbed into the error term, but the absorption should be stated explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: all central claims are derived from external theorems or direct character-sum computations; self-citations are motivational only.

full rationale

The paper's derivation chain is self-contained against external, independently published results. Theorem 1.1 is proved directly from Rudnick's lemmas and quadratic reciprocity, with no fitted parameters and no use of the conclusion as an input. Theorem 1.2 rests on the homological stability results of Bergström–Diaconu–Petersen–Westerland and Miller–Patzt–Petersen–Randal-Williams, which are cited as external theorems; the correction term -1/(q-1) emerges from Corollary 3.16, itself a computation from the stable-homology generating function, not from a fitted constant. Theorem 1.5 likewise uses the published Gauss-sum asymptotics of David–Florea–Lalín (DFL22), where one author overlaps with the present paper; however, that citation is to a prior established result that does not assume the target theorem and is not used to fit any parameter. The self-citations to HLOP25 and LOP25 are used only to motivate the murmuration phenomenon and are not load-bearing in any proof. The skeptic's concern about the non-decaying error term in Theorem 1.2 is a correctness risk about the stated hypotheses, not a circularity, because the error estimate is derived from bounds rather than being imposed as the answer. Overall, no prediction reduces by construction to its inputs, and no load-bearing uniqueness theorem is imported solely from the authors' own prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted free parameters are introduced; the stable range constants A and B are taken from a cited theorem, and the ε and σ in the cubic proof are standard analytic-number-theory tolerances rather than fitted values. No new particles, forces, or ad hoc entities are postulated.

assumptions (4)
  • standard math Uniform stable range H_k(β_n,V_λ) ≅ H_k(β_{n+1},V_λ) for k≤An+B with A=1/34, B=-35/34, cited from [MPPRW24, Proposition 1.5] and used as Theorem 3.3.
    This is the main topological input: it lets the authors replace finite-genus homology of H^{1,0}_g by stable homology up to degree θ(g). The whole correction-term computation in Section 3.4 depends on it.
  • standard math Stable homology generating function of H^{1,0}_∞ with symplectic irreducible coefficients and purity of these groups, taken from [BDPW23, Theorem 4.4.12 and Theorem 9.2.2], used as Theorem 3.12 and Theorem 3.8.
    Used to evaluate the hook contributions and to identify the n=2g correction through Corollary 3.16. If these computations failed, the cohomological description of the correction term would not follow.
  • standard math Asymptotic for averages of cubic Gauss sums, [DFL22, Proposition 3.1], used in the proof of Theorem 1.5 for all q≡1 mod 3.
    The Kummer cubic density result is derived almost entirely by substituting this asymptotic into the root number formulas. The theorem inherits the validity range of that proposition.
  • standard math Weil conjectures for curves over finite fields and the Katz-Sarnak model of USp(2g) monodromy, used to define Θ_D and to identify the random matrix baseline (3.22).
    Standard background in the field; it is used to interpret L(s,χ_D) as det(I - q^{1/2-s}Θ_D) and to set the expected random matrix trace in Theorem 1.2.

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Pith. "Pith review of Murmurations of quadratic and cubic characters over function fields." pith.science (2026). https://pith.science/paper/FI32PJMS

@misc{pith2026260801337,
  author       = {Pith},
  title        = {Pith review of: Murmurations of quadratic and cubic characters over function fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FI32PJMS}},
  note         = {Machine review of arXiv:2608.01337}
}
abstract

We compute the murmuration density for the family of quadratic characters over function fields. We show that the expectation of this density coincides with a correction term arising in the traces of high powers of the Frobenius class in this family, for which we determine large-genus asymptotics. This correction term admits a description in terms of the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point. We further identify this correction as a lower-order term in the one-level density and we obtain a non-vanishing result at the central point for quadratic $L$-functions over function fields, via an alternative approach to using the ratios conjecture. We also compute the murmuration density for the thin family of primitive cubic characters in the Kummer setting.

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Works this paper leans on

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