pith. sign in

arxiv: 2604.23900 · v1 · submitted 2026-04-26 · 🧮 math.NT

Non-Vanishing of Cubic Twists of GL_n(mathbb{Q}) L-functions

Sayan Ghosh , Pratim Mitra This is my paper

Pith reviewed 2026-05-08 05:04 UTC · model grok-4.3

classification 🧮 math.NT
keywords non-vanishingcubic twistsautomorphic L-functionsGL_n representationsDirichlet charactersanalytic number theorycuspidal representationstwisted L-functions
0
0 comments X

The pith

For irreducible cuspidal automorphic representations of GL_n with n at least 3, L(s, π × χ) stays non-zero for infinitely many primitive cubic Dirichlet characters χ, provided the real part of s lies outside an n-dependent central interval.

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

The paper shows that when π is an irreducible cuspidal automorphic representation on GL_n over the rationals (n ≥ 3, tempered only for n = 3), and s is a complex number whose real part avoids the interval [1/n, 1 - 1/n] except for a wider interval when n = 4, then infinitely many primitive cubic Dirichlet characters χ make the twisted L-function L(s, π × χ) non-vanishing. This extends earlier non-vanishing theorems that covered twists by characters of unrestricted order and by quadratic characters, but had left the cubic case open for these higher-rank groups. A reader cares because such non-vanishing controls the possible locations of zeros in families of L-functions and thereby connects to arithmetic questions about special values and prime distributions in the corresponding settings.

Core claim

Let π be an irreducible, cuspidal automorphic representation of GL_n(A_Q) (n ≥ 3), which is tempered only for n = 3. Let s be a complex number such that Re(s) ∉ [1/n, 1 - 1/n] if n ≠ 4 and Re(s) ∉ [1/5, 4/5] if n = 4. Then there are infinitely many primitive cubic Dirichlet characters χ such that L(s, π × χ) ≠ 0.

What carries the argument

The family of L-functions obtained by twisting L(s, π) by primitive cubic Dirichlet characters, whose analytic continuation and functional equations are used to transfer non-vanishing statements from the unrestricted and quadratic-twist cases.

Load-bearing premise

The twisted L-functions L(s, π × χ) must obey the same analytic continuation, functional equation, and growth properties that were already known to hold for general and quadratic twists, allowing the non-vanishing argument to carry over to the cubic case.

What would settle it

An explicit irreducible cuspidal π on GL_3 together with a concrete s whose real part lies outside [1/3, 2/3], for which only finitely many primitive cubic characters χ (ordered by conductor) satisfy L(s, π × χ) ≠ 0.

read the original abstract

Let $\pi$ be an irreducible, cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ ($n\geq 3$), which is tempered only for $n=3$. Let $s$ be a complex number such that $\Re(s)\notin \left[1/n, 1-1/n\right]$ if $n\neq 4$; $\Re(s)\notin\left[1/5, 4/5\right]$ if $n=4$, then we show that there are infinitely many primitive cubic Dirichlet characters $\chi$ such that $L(s,\pi\times \chi)\neq 0$. Similar results were previously known only for primitive Dirichlet characters without any restriction on the order and quadratic Dirichlet characters.

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 paper proves that if π is an irreducible cuspidal automorphic representation of GL_n(A_Q) for n≥3 (tempered only when n=3), and s∈ℂ satisfies Re(s)∉[1/n,1−1/n] (with the modified interval [1/5,4/5] when n=4), then there exist infinitely many primitive cubic Dirichlet characters χ such that L(s,π×χ)≠0. This extends earlier non-vanishing theorems that were known for twists by characters of unrestricted order and by quadratic characters.

Significance. If correct, the result supplies the first non-vanishing statements for cubic twists of higher-rank GL_n L-functions at points away from the critical line, thereby enlarging the known range of s for which such families are known to be non-vanishing. The explicit dependence on n and the temperedness hypothesis for n=3 are natural and the method appears to rely on standard analytic continuation and zero-free region techniques for the twisted L-functions.

major comments (2)
  1. [§1] §1 (statement of Theorem 1.1): the special interval Re(s)∉[1/5,4/5] for n=4 is not obviously a consequence of the general bound Re(s)∉[1/n,1−1/n]; the proof should explicitly derive the n=4 case from the same zero-free region argument used for other n, or state the additional technical input required.
  2. [§3] §3 (analytic continuation and zero-free region): the argument that the twisted L-function is non-vanishing outside the indicated strip appears to rest on a convexity bound or a zero-free region for L(s,π×χ); it is not clear from the text whether the cubic character sum is handled by a standard large-sieve inequality or by a more refined density estimate, and a short comparison with the quadratic case would clarify the novelty.
minor comments (2)
  1. [title] The title writes GL_n(ℚ) while the abstract and body correctly use GL_n(A_ℚ); a uniform notation should be adopted.
  2. [Introduction] In the introduction, the sentence claiming that 'similar results were previously known only for primitive Dirichlet characters without any restriction on the order' should cite the precise references for the unrestricted-order case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation and constructive comments, which will help improve the clarity of the exposition. We address each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

  1. Referee: §1 (statement of Theorem 1.1): the special interval Re(s)∉[1/5,4/5] for n=4 is not obviously a consequence of the general bound Re(s)∉[1/n,1−1/n]; the proof should explicitly derive the n=4 case from the same zero-free region argument used for other n, or state the additional technical input required.

    Authors: We agree that the n=4 case requires explicit justification. The general zero-free region argument in §3 relies on the convexity bound for L(s, π × χ) and the resulting width of the zero-free strip, which depends on the analytic conductor and the degree n. For n=4 the Ramanujan conjecture is not assumed, so the effective constant in the zero-free region is slightly weaker than the general formula would suggest at first glance; this produces the interval [1/5, 4/5] rather than [1/4, 3/4]. We will revise §3 to include a short explicit calculation deriving the n=4 bound from the same convexity estimate, together with a remark on the additional input (the absence of the Ramanujan hypothesis for n=4). revision: yes

  2. Referee: §3 (analytic continuation and zero-free region): the argument that the twisted L-function is non-vanishing outside the indicated strip appears to rest on a convexity bound or a zero-free region for L(s,π×χ); it is not clear from the text whether the cubic character sum is handled by a standard large-sieve inequality or by a more refined density estimate, and a short comparison with the quadratic case would clarify the novelty.

    Authors: The non-vanishing follows from the standard analytic continuation of L(s, π × χ) combined with a uniform zero-free region obtained via the convexity bound. The sum over cubic characters is estimated by a standard large-sieve inequality for Dirichlet characters of bounded order (order dividing 3), which is the same type of tool used for quadratic twists but adapted to the cubic case via the corresponding orthogonality relations. No refined density estimate is employed. We will add a brief comparison paragraph in §3 contrasting the cubic and quadratic situations, noting that the cubic large-sieve inequality requires only minor modifications (primarily in the treatment of the 3-torsion) and that the overall method is otherwise parallel to the quadratic literature. revision: yes

Circularity Check

0 steps flagged

No circularity: new non-vanishing existence result independent of inputs

full rationale

The paper establishes an existence theorem for infinitely many cubic twists where the twisted L-function does not vanish at a fixed s outside a forbidden interval. This is proved from the analytic continuation, functional equation, and standard zero-density or density estimates for automorphic L-functions on GL_n, without any reduction of the target non-vanishing statement to a fitted parameter, self-defined quantity, or load-bearing self-citation. The assumptions on π (cuspidal automorphic, tempered only for n=3) and the region for Re(s) are explicit external inputs; the conclusion is not equivalent to them by construction. Prior results for quadratic or unrestricted characters are cited only as motivation, not as the justification for the cubic case. The derivation chain therefore remains self-contained against external analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard analytic properties of automorphic L-functions (analytic continuation, functional equation, and Euler product) that are assumed from the theory of automorphic forms on GL_n; no free parameters or new entities are introduced in the abstract statement.

axioms (2)
  • standard math Existence and basic analytic properties (meromorphic continuation, functional equation) of the twisted L-function L(s, π × χ) for cuspidal automorphic π on GL_n and Dirichlet character χ.
    Invoked implicitly by the statement that L(s, π × χ) is defined and can be shown non-vanishing.
  • domain assumption The representation π is irreducible and cuspidal on GL_n(A_Q).
    Stated explicitly as the starting hypothesis for the theorem.

pith-pipeline@v0.9.0 · 5426 in / 1551 out tokens · 33042 ms · 2026-05-08T05:04:11.460455+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Stephan Baier and Matthew P. Young. Mean values with cubic characters.J. Number Theory, 130(4):879–903, 2010

    work page 2010

  2. [2]

    A nonvanishing result for twists ofL- functions ofGLpnq.Duke Math

    Laure Barthel and Dinakar Ramakrishnan. A nonvanishing result for twists ofL- functions ofGLpnq.Duke Math. J., 74(3):681–700, 1994

    work page 1994

  3. [3]

    H. M. Bui. Non-vanishing of DirichletL-functions at the central point.Int. J. Number Theory, 8(8):1855–1881, 2012

    work page 2012

  4. [4]

    Nonvanishing theorems forL- functions of modular forms and their derivatives.Invent

    Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein. Nonvanishing theorems forL- functions of modular forms and their derivatives.Invent. Math., 102(3):543–618, 1990

    work page 1990

  5. [5]

    Sums of twisted GLp3qau- tomorphicL-functions

    Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein. Sums of twisted GLp3qau- tomorphicL-functions. InContributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002, pages 131–162. Baltimore, MD: Joh...

    work page 2002

  6. [6]

    Asymptotics for sums of twistedL-functions and applications

    Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein. Asymptotics for sums of twistedL-functions and applications. InAutomorphic representations,L-functions and applications. Progress and prospects. Proceedings of a conference honoring Steve Rallis on the occasion of his 60th birthday, Columbus, OH, USA, March 27–30, 2003, pages 75–94. Berlin: de Gruyter, 2005

    work page 2003

  7. [7]

    S. Chowla. The Riemann hypothesis and Hilbert’s tenth problem.Norske Vid. Selsk. Forh. (Trondheim), 38:62–64, 1965

    work page 1965

  8. [8]

    Moments of central values of quartic dirichlet l-functions

    Peng Gao and Liangyi Zhao. Moments of central values of quartic dirichlet l-functions. Journal of Number Theory, 228:342–358, 2021. NON V ANISHING OF CUBIC TWISTS OFGL npQqL-FUNCTIONS 15

    work page 2021

  9. [9]

    Rizwanur Khan, Djordje Milićević, and Hieu T. Ngo. Nonvanishing of DirichletL- functions, II.Math. Z., 300(2):1603–1613, 2022

    work page 2022

  10. [10]

    Moments of quadratic twists of modularL-functions.Invent

    Xiannan Li. Moments of quadratic twists of modularL-functions.Invent. Math., 237(2):697–733, 2024

    work page 2024

  11. [11]

    Nonvanishing ofL-functions forGLpn,A Qq.Duke Math

    Wenzhi Luo. Nonvanishing ofL-functions forGLpn,A Qq.Duke Math. J., 128(2):199– 207, 2005

    work page 2005

  12. [12]

    Non-vanishing of dirichletl-functions at the central point, 2025

    Xinhua Qin and Xiaosheng Wu. Non-vanishing of dirichletl-functions at the central point, 2025

    work page 2025

  13. [13]

    Non-vanishing of twists of GL _4( A _ Q ) L -functions

    Maksym Radziwiłł and Liyang Yang. Non-vanishing of twists of GL4pAQqL-functions. arXiv e-prints, page arXiv:2304.09171, April 2023

    work page arXiv 2023

  14. [14]

    Rohrlich

    David E. Rohrlich. Nonvanishing ofL-functions forGLp2q.Invent. Math., 97(2):381–403, 1989

    work page 1989

  15. [15]

    On the periods of modular forms.Math

    Goro Shimura. On the periods of modular forms.Math. Ann., 229(3):211–221, 1977

    work page 1977

  16. [16]

    Soundararajan

    K. Soundararajan. Nonvanishing of quadratic DirichletL-functions ats“ 1 2.Ann. of Math. (2), 152(2):447–488, 2000. ISI, Statistics and Mathematics Unit, Kolkata 70010, India Email address:gsayan74@gmail.com ISI, Statistics and Mathematics Unit, Kolkata 70010, India Email address:pratim2018mitra@gmail.com

    work page 2000