Nonvanishing of $L$--functions associated to fixed order characters over function fields

Alexandra Florea; Chantal David; Matilde Lalin

REVIEW 3 major objections 5 minor 1 cited by

Nonvanishing of $L$--functions associated to fixed order characters over function fields

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Positive share of order-$\ell$ $L$-functions avoid vanishing at the central point

desk verdict Genuinely new positive-proportion nonvanishing for arbitrary fixed order ℓ in function fields; substantial but needs referee attention to a generating-series definition mismatch. read the letter →

arxiv 2506.07815 v1 pith:H75EZBLE submitted 2025-06-09 math.NT

Chantal David , Alexandra Florea , Matilde Lalin This is my paper · ORCID

classification math.NT MSC 11M0611M3811R1611R58

keywords functionfieldsnon-vanishingone-leveldensitylow-lyingzerosfixedordercharactersGausssumslargesieveL-functions

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

The paper proves that, for every fixed order $\ell\ge 3$ with $q\equiv 1 \pmod{2\ell}$ and $\ell\nmid \deg(c)$, a positive proportion of the $L$-functions attached to the $\ell$-th residue characters over $\mathbb{F}_q[t]$ do not vanish at $s=1/2$ as $\deg(c)\to\infty$. This upgrades the previous infinitude statement for general $\ell$ to an explicit positive fraction of the family $F_\ell(d)$. The proof uses a one-level density computation for the low-lying zeros, pushing the Fourier support of the test function beyond the interval $(-1,1)$, which is exactly what forces a positive proportion of nonvanishing for unitary families. The stated fractions are $1/6$ for cubic characters, $3/26$ for quartic characters, and smaller explicit positive fractions for every $\ell\ge 5$.

What carries the argument

The load-bearing object is the generating series $\psi^{(i)}(r,u)=(1-q^\ell u^\ell)^{-1}\sum_{\deg F\equiv i\pmod{\ell}} G_\ell(r,F)u^{\deg F}$, which packages shifted order-$\ell$ Gauss sums by degree class. The paper combines the rationality and functional equation of these series with a new approximate functional equation for their numerators, then controls their average size with large-sieve inequalities for order-$\ell$ characters and a Vaughan-identity decomposition of the sums over primes. Beating the pointwise convexity bound on these averages is what pushes the one-level density support past the $(-1,1)$ barrier.

What would settle it

A direct check would be to run the large-sieve inequality of Theorem 1.4 on an explicit sequence of coefficients $\lambda(N)$: a choice of supports $m,n$ that violates the stated $q^{2(m+n)/3}$ term would collapse the Type II bound. A complementary check is to enumerate $F_\ell(d)$ for a fixed small $\ell$ and growing $d$ and compare the nonvanishing proportion against the Corollary 1.2 fraction, though only a persistent violation as $d$ grows would be decisive.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for the family $F_\ell(d)$ of order-$\ell$ residue characters with square-free modulus of degree $d$, the one-level density of zeros equals $\widehat{\varphi}(0)+O(1/d)$ for even Schwartz test functions whose Fourier transform is supported in $(-v,v)$, with $v$ strictly larger than $1$ and given piecewise: $v=6/5$ for $\ell=3$, $v=26/23$ for $\ell=4$, and slightly larger than $1$ for $\ell\ge 5$. Because the symmetry type is unitary, the condition $v>1$ is precisely what lets the density force a zero at the central point, and Corollary 1.2 converts this into the nonvanishing proportions. As a by-product, Theorem 1.6 establishes cancellation in averages of shifted order-$\ell$ Gauss sums $G_\ell(R,\pi)$ over primes $\pi$, uniformly in the shift $R$, which the paper presents as a step toward equidistribution of their angles.

Load-bearing premise

The load-bearing premise is that the imported bounds for the generating series of shifted Gauss sums are exactly as stated in Theorem 4.1; if those bounds were weaker, the error terms would not be small enough to push the one-level density support beyond 1, and the positive proportion would not follow.

Editorial extensions

If this is right

For every $\ell\ge 3$ there are positive proportions of nonvanishing central values in the family $F_\ell(d)$, with the explicit fractions $1/6$ for $\ell=3$, $3/26$ for $\ell=4$, and $2(\ell-2)/(2\ell^2+\ell-2)$, $2(\ell-2)/(3\ell^2-7\ell-2)$, or $6(\ell-2)/(9\ell^2-25\ell-6)$ in the stated ranges of $\ell$.
The one-level density matches the unitary prediction $\widehat{\varphi}(0)$ up to $O(1/d)$, for Fourier support reaching $v>1$ in every case considered.
Averaged shifted order-$\ell$ Gauss sums over primes have cancellation bounds that are uniform in the shift $R$, a feature not available for general $\ell$ in the number-field setting.
The unconditional large-sieve inequalities for order-$\ell$ characters are proved as standalone theorems and can be used in other families.

Reading between the lines

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

Looking beyond the paper, any strengthening of the Lindelöf-on-average bounds for $\psi^{(i)}$ would enlarge the allowed support $v$ and thereby raise the nonvanishing fractions; the authors explicitly note they did not optimize small $\ell$ cases to keep the paper shorter.
Theorem 1.6 stops short of full equidistribution of the angles of $G_\ell(R,\pi)$ because only a restricted class of generating averages is bounded; extending the argument to higher moments would supply Weyl-criterion equidistribution with uniformity in $R$.
Since the Corollary 1.2 fractions decay like $O(1/\ell)$ as $\ell$ grows, the method does not address Chowla-type nonvanishing of every character; the paper itself remarks that the proportion approaches zero as $\ell\to\infty$.
The same template—Vaughan's identity, Poisson summation, and the order-$\ell$ large sieve—should apply to other thin families of fixed-order characters in function fields, such as non-Kummer settings where the base field lacks the $\ell$-th roots of unity.

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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

First thing you should know: this is the first positive-proportion nonvanishing result for L(1/2, χ_c) where χ_c is an order-ℓ character over F_q[t] for arbitrary fixed ℓ≥3. Previous positive proportions only covered ℓ=2,3,4; general ℓ only had infinitely many nonvanishing twists (ELS20). The authors also get support v>1 in the one-level density, a new large sieve for order-ℓ characters, and a cancellation bound for shifted Gauss sums at primes. The main theorem has explicit proportions (1/6 for ℓ=3, etc.). All of that is real.

The paper is long and carefully structured: approximate functional equation, Vaughan's identity, Type I/II sums, large sieve. The proofs are mostly there, and the external inputs are standard or published. I believe the central argument is sound at the structural level.

The soft spots, in order of real softness. First, there is a genuine mismatch in Section 4: ψ^(i)(r,u) is defined with denominator (1 - q^ℓ u^ℓ), but the functional equation and equation (4.2) use (1 - q^{ℓ+1} u^ℓ). Those are different objects; the first has a pole at u^ℓ = q^{-ℓ}, the second at q^{-ℓ-1}. This is probably a typo in the definition, but it is load-bearing: Theorem 4.1, the approximate functional equation, and everything after depends on the q^{ℓ+1} version. A referee must ask the authors to fix it and confirm the residue statements. Second, Theorem 4.1 is only sketched, with the messy case analysis imported from [DFL22], which is written for ℓ=3. The generalization to all ℓ is plausible but not fully shown in this preprint. Third, §7.1 invokes the Lindelöf hypothesis without saying it is the known function-field theorem; that's a clarity issue.

None of these, on the evidence here, overturns the result. But they are exactly why the right verdict is conditional, not unconditional. The paper deserves a serious referee: it is important, the techniques are novel, and the gaps are addressable. I would recommend sending it out with a request to fix the generating-series definition and to either prove Theorem 4.1 in the generality claimed or give a precise pointer. If those checks pass, the result stands.

Referee Report

3 major / 5 minor

Summary. The paper studies the family F_ℓ(d) of order-ℓ Dirichlet characters over F_q[t] in the Kummer setting q≡1 mod 2ℓ with ℓ∤d. Theorem 1.1 computes the one-level density of low-lying zeros for this family, with test functions whose Fourier transform is supported in (-v,v) for a piecewise-defined v>1; Corollary 1.2 converts this into a positive proportion of nonvanishing central values L(1/2,χ_c), for example at least 1/6 for ℓ=3 and 3/26 for ℓ=4. The proof uses the explicit formula, Poisson summation, Vaughan's identity, a combinatorial decomposition of generating series of Gauss sums (Theorem 5.3), Lindelöf-on-average bounds for these series (Section 6), large-sieve inequalities (Theorems 1.4 and 1.5), and a Type I/Type II analysis. The paper also proves Theorem 1.6, a cancellation bound for averaged shifted Gauss sums at prime arguments.

Significance. If the analytic ingredients are valid, this is a substantial advance: it gives the first positive-proportion nonvanishing result for fixed-order characters of arbitrary order ℓ in the function-field Kummer setting, improving on the infinitely-many result of Ellenberg–Li–Shusterman. The proof is highly structured and includes useful standalone tools, in particular large-sieve inequalities for order-ℓ characters and a uniform Gauss-sum cancellation result. The main risk is the imported analytic theory for the generating series ψ^(i): the paper relies on published work of Hoffstein, Patterson, and the authors' own ℓ=3 paper, and the exact form of the required statements and exponents is not fully pinned down in the present text. These issues are fixable but load-bearing.

major comments (3)

[§4, Eqs. (4.1)–(4.4), Theorems 4.1 and 4.4] The function defined at the start of §4 is ψ(i)(r,u)=(1−q^ℓ u^ℓ)^{-1}Σ..., while Eq. (4.1), Eq. (4.2), and Theorem 4.4 concern an object with denominator 1−q^{ℓ+1}u^ℓ and poles at u^ℓ=q^{−ℓ−1}. The manuscript never explains the cancellation that reconciles these two objects; without it, the convexity bound in Theorem 4.1 on q^{−3/2}≤|u|≤q^{−1/2}, away only from u^ℓ=q^{−ℓ−1}, would fail at the pole u^ℓ=q^{−ℓ} of the displayed definition. The proof of Theorem 4.1 is also only a sketch: the case analysis for i≠j is delegated to [DFL22], which treats ℓ=3, and the displayed relation for ψ(j)(r,s) has ψ(j) on both sides (the second term should be ψ(i)(r,2−s)). Since the v>1 support in Theorem 1.1 is inherited from the Lindelöf-on-average bounds of Section 6, which use Theorems 4.1 and 4.4, this point must be stated precisely and proved completely.
[§8, Eq. (8.3); §12, Lemma 12.2 and Theorem 1.4] Theorem 1.4 is stated only for the character (M/N)_ℓ, but the Type II application in (8.2)–(8.3) requires the same bound for (α/c)^2_ℓ, and Lemma 12.2 requires it for (M_k/N)^k_ℓ for each 1≤k≤ℓ−1. For even ℓ, χ^2 has order ℓ/2, so the statement as written does not apply; this matters already for ℓ=4. The proof of Theorem 1.4 appears to go through unchanged for any fixed k, so please state and prove the generalized large sieve for all nontrivial powers, or supply the additional quadratic/cubic large-sieve input needed for even ℓ.
[§5, Theorem 5.3] The denominator product in Theorem 5.3 is not consistent with the factors derived in the proof. In the step-by-step derivation, a prime π|r_{ℓ−1} contributes a factor with exponent 1 and no (−1/π)_ℓ symbol, while a prime π|aE contributes exponent ℓ−1; the displayed product over π|aEr_1...r_{ℓ−1} with exponent ν_π(aEr_1...r_{ℓ−2}r_{ℓ−1})+1 gives exponent 2 for π|r_{ℓ−1} and, typically, 2 or 3 for π|aE. Please correct the formula, or state explicitly that only the O(1) size of these factors is used in the later bounds. Since Theorem 5.3 is the pivot for Corollary 5.4, Proposition 6.3, and the Type I bounds, the exact statement must be reliable.

minor comments (5)

[§7.1] The phrase 'using the Lindelöf hypothesis' for the bound Σ_{c∈H_d} χ_c(f)≪q^{d/2+εd} should be replaced by a precise reference to the relevant Weil/Riemann-hypothesis bound in function fields; as written, it sounds like an unproved assumption, even though the paper's results are meant to be unconditional.
[§4, proof of Theorem 4.1] In the displayed formula for ψ(j)(r,s), the second term on the right-hand side repeats ψ(j)(r,2−s); it should be ψ(i)(r,2−s), otherwise the displayed identity is tautological.
[§5, Lemma 5.1(5.3)] The range '1≤j≤n−2' should be '1≤j≤ℓ−2'; the letter n is not defined in that lemma.
[§10.1] After Eq. (7.2), the notation F_ℓ(g) should be F_ℓ(d).
[§6, Remark 6.5] The comparison statement in Remark 6.5 says 'for ℓ≥8' but the condition in Corollary 5.4 at σ=1+1/ℓ appears to switch cases at ℓ>8; please check whether ℓ=8 belongs to the first or second case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: main derivation is self-contained; imported generating-series facts are external prior theorems, not restatements of the target.

full rationale

Walking the derivation chain, Theorem 1.1 is obtained from the explicit formula (Section 7.1), Poisson summation (Lemma 3.3), Vaughan decomposition (Section 7.2), Type II bounds via the large sieve (Section 8, with Theorems 1.4 and 1.5 proved independently in Section 12), Type I bounds via generating series (Section 9), and the Lindelof-on-average bounds of Section 6. None of these steps is definitionally equal to the target: the one-level density is not assumed but computed from character sums, and the support threshold v is balanced analytically from the error terms, with no fitted parameter renamed as a prediction. The generating series psi^(i)(r,u) is defined in Section 4 directly from the order-l Gauss sums of Section 3, and its rationality, functional equation, and convexity bound are imported from Hof92, Pat07, and the authors' earlier DFL22 paper. These are independent published results: DFL22 establishes the analogous generating-series facts for l=3 and does not contain the present nonvanishing theorem or one-level density computation, so the self-citation is real evidence rather than a circular premise. The skeptical observation that Section 4 defines psi with denominator 1-q^l u^l while equation (4.2) displays denominator 1-q^{l+1}u^l is an expositional or correctness risk in the imported technical toolbox, not a reduction of the paper's conclusion to its input. Likewise, the delegation of case analysis in the sketch of Theorem 4.1 to DFL22 may be a completeness concern for general l, but it is not a case where the claimed result is equivalent by construction to the cited prior work. I therefore find no circular step under any of the seven patterns; the correct score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim is a theorem in analytic number theory; it has no empirical free parameters. The auxiliary cutoffs U(n), y, and v are explicit choices that optimize error terms, not fitted values. The main axioms are standard function-field results (reciprocity, Weil bound, Pólya-Vinogradov, Riemann-Hurwitz) plus the theory of metaplectic generating series of higher-order Gauss sums developed in Hoffstein, Patterson, and the authors' earlier work [DFL22]. No new physical or mathematical entities are postulated.

free parameters (3)

Vaughan identity cutoff U(n) = piecewise: n/3 for ℓ=3; 3n/13 for ℓ=4; (n+d)(ℓ-2)/ℓ² for 5≤ℓ≤8; similar explicit formulas for ℓ≥9
Auxiliary cutoff balancing Type I and Type II sums; chosen by hand in Sections 8-10. The final theorem holds for all large d, so this is an optimization parameter, not a data-fitted constant.
Möbius cutoff y = 100εd
Auxiliary cutoff on the D-sum in Section 7; chosen small to ensure an error term q^{-εd}; not fitted to data.
Fourier support threshold v = 6/5, 26/23, and formulas in Theorem 1.1 for ℓ≥5
The largest support for which the one-level density error terms are negligible; part of the theorem's content rather than a fitted constant, but a hand-optimized boundary.

assumptions (5)

domain assumption ℓ-th power reciprocity law for F_q[t] with q≡1 (mod 2ℓ) (Section 2).
Used to replace χ_c(π) by χ_π(c) when swapping summations; the Kummer condition q≡1 mod 2ℓ makes the reciprocity sign trivial.
standard math Weil bound for character sums over primes and function-field RH for order-ℓ L-functions (equation (2.8)).
Controls prime sums in the explicit formula; standard consequence of Weil's proof of RH for function fields, cited via Rudnick.
standard math Pólya-Vinogradov bound in function fields (equation (2.9), Hsu99).
Bounds incomplete character sums after Poisson summation; external published result.
domain assumption Rationality, functional equation, and convexity/residue bounds for the generating series ψ^(i)(r,u) (Section 4, equations (4.1)-(4.4), Theorem 4.1).
Imported from Hoffstein, Patterson, and DFL22; this is the main external input and the least standard background assumption.
standard math Riemann-Hurwitz genus formula and degree bound deg L(u,χ_c)=d-1 for odd χ_c (Section 2, equation (2.2)).
Ensures the L-function is a polynomial of degree d-1 so the explicit formula has d-1 zeros; standard in the function field literature.

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Pith. "Pith review of Nonvanishing of $L$--functions associated to fixed order characters over function fields." pith.science (2026). https://pith.science/paper/H75EZBLE

@misc{pith2026250607815,
  author       = {Pith},
  title        = {Pith review of: Nonvanishing of $L$--functions associated to fixed order characters over function fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H75EZBLE}},
  note         = {Machine review of arXiv:2506.07815}
}

abstract

We show that a positive proportion of the values $L(1/2,\chi_c)$ are non-zero, where $\chi_c$ is the $\ell^{\text{th}}$ residue symbol for $\ell \geq 3$ over $\mathbb{F}_q[t]$, when averaging over square-free polynomials $c$ in $\mathbb{F}_q[t]$, as $q \equiv 1(\textrm{mod}\,{2\ell})$ is fixed and the degree of $c$ goes to infinity. In the case of $\ell=3$, we show that at least $1/6$ of $L(1/2,\chi_c)\neq 0$, while for $\ell>3$, the proportion depends on the order of the character. This improves a previous result of Ellenberg, Li, and Shusterman showing that there are infinitely many $\chi$ of (prime) order $\ell$ such that $L(1/2, \chi) \neq 0$ (with completely different techniques). Our result is achieved by computing the one-level density of zeros in the family of $L$--functions and surpassing the $(-1,1)$ barrier for the support of the Fourier transform of the test function, necessary to obtain a positive proportion of non-vanishing result. Using similar techniques, we also prove a result towards the equidistribution of the angles of the order $\ell$ shifted Gauss sums when summing over prime arguments, a result which may be of independent interest.

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Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$
math.NT 2026-08 conditional novelty 7.0 of 10

Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

H. M. Bui and Alexandra Florea, Zeros of quadratic D irichlet L -functions in the hyperelliptic ensemble , Trans. Amer. Math. Soc. 370 (2018), no. 11, 8013--8045. 3852456

work page 2018

[2] [2]

Valentin Blomer, Leo Goldmakher, and Beno\^it Louvel, L -functions with n -th-order twists , Int. Math. Res. Not. IMRN (2014), no. 7, 1925--1955. 3190355

work page 2014

[3] [3]

H. M. Bui, Non-vanishing of D irichlet L -functions at the central point , Int. J. Number Theory 8 (2012), no. 8, 1855--1881. 2978845

work page 2012

[4] [4]

Young, Mean values with cubic characters, J

Stephan Baier and Matthew P. Young, Mean values with cubic characters, J. Number Theory 130 (2010), no. 4, 879--903. 2600408

work page 2010

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Number Theory 8 (2022), no

Antoine Comeau-Lapointe, Chantal David, Matilde Lalin, and Wanlin Li, On the vanishing of twisted L -functions of elliptic curves over rational function fields , Res. Number Theory 8 (2022), no. 4, Paper No. 76, 28. 4491491

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Chantal David, Alexandre de Faveri, Alexander Dunn, and Joshua Stucky, Non-vanishing for cubic Hecke L -functions , arXiv e-prints, arXiv:2410.03048

arXiv

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Chantal David, Alexander Dunn, Alia Hamieh, and Hua Lin, Quartic gauss sums over primes and metaplectic theta functions, arXiv e-prints, arXiv:2306.11875

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Sigma 9 (2021), Paper No

Chantal David, Alexandra Florea, and Matilde Lalin, Nonvanishing for cubic L -functions , Forum Math. Sigma 9 (2021), Paper No. e69, 58. 4323990

work page 2021