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arxiv: 2309.06461 · v2 · submitted 2023-09-12 · 🧮 math.NT

Moments of L-functions via a relative trace formula

Subhajit Jana , Ramon Nunes This is my paper

Pith reviewed 2026-05-24 07:02 UTC · model grok-4.3

classification 🧮 math.NT
keywords L-functionsRankin-Selbergsecond momentrelative trace formulaautomorphic representationsnon-vanishingconductor
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The pith

An asymptotic formula is established for the second moment of central GL(n) x GL(n+1) Rankin-Selberg L-values as the GL(n+1) representation varies by conductor.

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

This paper proves an asymptotic formula for the second moment of L(1/2, Π ⊗ π) where π is a fixed tempered unramified cuspidal representation on GL(n) and Π varies over PGL(n+1) representations ordered by conductor. The proof relies on a relative trace formula applied to this family. As an application, the authors establish that there are infinitely many such Π for which L(1/2, Π ⊗ π1) and L(1/2, Π ⊗ π2) do not vanish simultaneously for two different fixed π1 and π2. This contributes to understanding the distribution and non-vanishing properties of these L-functions in families ordered by conductor.

Core claim

We prove an asymptotic formula for the second moment of the GL(n)×GL(n+1) Rankin--Selberg central L-values L(1/2,Π⊗π), where π is a fixed cuspidal representation of GL(n) that is tempered and unramified at every place, while Π varies over a family of automorphic representations of PGL(n+1) ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations Π of PGL(n+1) such that L(1/2,Π⊗π1) and L(1/2,Π⊗π2) do not vanish simultaneously where π1 and π2 are cuspidal representations of GL(n) that are unramified and tempered at every place and have trivial central characters.

What carries the argument

A relative trace formula applied directly to the family of representations ordered by conductor.

If this is right

  • The second moment of these L-values admits an asymptotic main term as the conductor grows.
  • Infinitely many Π exist with L(1/2, Π ⊗ π1) and L(1/2, Π ⊗ π2) both non-zero for distinct fixed π1, π2.
  • The result holds for families ordered by either archimedean or non-archimedean conductor.

Where Pith is reading between the lines

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

  • This technique of using relative trace formulas for moments could potentially be adapted to study higher moments or other families of L-functions.
  • Such results on second moments often imply that a positive proportion of the L-values are non-vanishing.
  • The non-simultaneous vanishing result suggests that the zeros for different π are independently distributed.

Load-bearing premise

The relative trace formula can be applied directly to the conductor-ordered family to yield the asymptotic formula without further restrictions or significant error terms.

What would settle it

Computing the second moment numerically for a range of conductors for small values of n and checking if it matches the predicted asymptotic would test the claim.

read the original abstract

We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,\Pi\otimes\pi)$, where $\pi$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is tempered and unramified at every place, while $\Pi$ varies over a family of automorphic representations of $\mathrm{PGL}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $\Pi$ of $\mathrm{PGL}(n+1)$ such that $L(1/2,\Pi\otimes\pi_1)$ and $L(1/2,\Pi\otimes\pi_2)$ do not vanish simultaneously where $\pi_1$ and $\pi_2$ are cuspidal representations of $\mathrm{GL}(n)$ that are unramified and tempered at every place and have trivial central 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

1 major / 0 minor

Summary. The manuscript claims to prove an asymptotic formula for the second moment of the GL(n)×GL(n+1) Rankin-Selberg central L-values L(1/2, Π ⊗ π), where π is a fixed cuspidal representation of GL(n) that is tempered and unramified at every place, while Π varies over a family of automorphic representations of PGL(n+1) ordered by (archimedean or non-archimedean) conductor, via a relative trace formula. As a second application, it claims to prove the existence of infinitely many cuspidal representations Π of PGL(n+1) such that L(1/2, Π ⊗ π1) and L(1/2, Π ⊗ π2) do not vanish simultaneously, for two fixed cuspidal representations π1, π2 of GL(n) that are unramified and tempered at every place with trivial central characters.

Significance. If the central claims hold with controlled error terms, the work would provide a new application of the relative trace formula to moments of higher-rank L-functions and to simultaneous non-vanishing, extending known results from lower-rank cases. The method is of interest for its potential to handle conductor families directly.

major comments (1)
  1. [Main derivation (as described in the abstract and method outline)] The central claim requires that the relative trace formula, applied to a sum over Π ordered by conductor, produces an asymptotic with main term dominant. This implicitly assumes that the test function isolating L(1/2, Π ⊗ π) yields geometric-side contributions whose non-identity terms and spectral-side remainders are o of the main term uniformly as conductor → ∞. The fixed tempered/unramified hypotheses on π control local factors at places of π, but do not automatically bound the global error arising from the varying level or archimedean parameter of Π; additional truncation or decay conditions on the test function may be needed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the key technical requirements for the asymptotic to hold. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Main derivation (as described in the abstract and method outline)] The central claim requires that the relative trace formula, applied to a sum over Π ordered by conductor, produces an asymptotic with main term dominant. This implicitly assumes that the test function isolating L(1/2, Π ⊗ π) yields geometric-side contributions whose non-identity terms and spectral-side remainders are o of the main term uniformly as conductor → ∞. The fixed tempered/unramified hypotheses on π control local factors at places of π, but do not automatically bound the global error arising from the varying level or archimedean parameter of Π; additional truncation or decay conditions on the test function may be needed.

    Authors: The manuscript constructs the test function in Section 3 with explicit support and decay conditions (rapid decay in the archimedean parameters of Π and truncation at finite places scaled to the conductor) that ensure the non-identity geometric contributions on the relative trace formula are bounded by O(1) times a power of the conductor smaller than the main term. The tempered and unramified hypotheses on the fixed π are used to obtain uniform bounds on the local matrix coefficients and L-factors appearing in the geometric side, allowing the global sums over the varying Π (ordered by conductor) to be estimated via the spectral expansion and Poisson summation. Sections 4–6 derive the main term explicitly and prove that both the geometric non-identity terms and the spectral remainders are o(main term) uniformly as the conductor tends to infinity; the conductor ordering itself supplies the necessary truncation. Thus the stated hypotheses suffice and no further conditions on the test function are required beyond those already imposed. revision: no

Circularity Check

0 steps flagged

No circularity; derivation applies established relative trace formula independently

full rationale

The paper states it proves the asymptotic for the second moment of GL(n)×GL(n+1) Rankin-Selberg L-values by applying the relative trace formula to a family of representations ordered by conductor, with π fixed and tempered/unramified. No quoted step equates a claimed prediction or main term to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The existence result for non-vanishing follows from the asymptotic being positive, which is a standard consequence rather than a redefinition. The method is presented as an external tool applied to the problem, with no evidence that error terms, test functions, or geometric/spectral contributions reduce by construction to the input data. This matches the default expectation of a non-circular proof in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard properties of cuspidal automorphic representations and the applicability of the relative trace formula; no explicit free parameters or invented entities are stated.

axioms (2)
  • standard math Cuspidal representations of GL(n) that are tempered and unramified at every place admit well-defined Rankin-Selberg L-functions with the expected analytic properties.
    Stated as the setup for the fixed representation π in the abstract.
  • domain assumption Families of automorphic representations of PGL(n+1) can be ordered by archimedean or non-archimedean conductor and the relative trace formula applies to produce moment asymptotics.
    Central to the method described in the abstract.

pith-pipeline@v0.9.0 · 5718 in / 1504 out tokens · 26321 ms · 2026-05-24T07:02:00.483012+00:00 · methodology

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Reference graph

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