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arxiv: 2407.05583 · v2 · submitted 2024-07-08 · 🧮 math.NT

The Ranking-Selberg integral on {bf GSp(2)} for square free levels

Seiji Kuga , Masao Tsuzuki This is my paper

Pith reviewed 2026-05-23 23:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords Rankin-Selberg integralGSp(2)Siegel cusp formssquare-free levelsBessel modelszeta-integralsautomorphic representations
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The pith

The Rankin-Selberg integral for vector-valued Siegel cusp forms on GSp(2) is computed explicitly when levels are square-free.

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

The paper establishes an explicit computation of the Rankin-Selberg type integral over the adeles for vector-valued Siegel cusp forms of square-free levels. This integral, introduced by Piatetski-Shapiro, factors into local zeta-integrals that receive exact evaluations for chosen test functions in the Bessel models of irreducible admissible representations. A sympathetic reader cares because such computations link global automorphic integrals to local data. The square-free condition enables the clean factorization described.

Core claim

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels Γ₀(N). On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.

What carries the argument

The adelic Rankin-Selberg integral that factors into local zeta-integrals evaluated at Bessel model test functions.

If this is right

  • The global integral equals a product of local zeta-integrals.
  • Exact closed-form expressions exist for the local integrals at the chosen test functions.
  • The result applies to square-free levels Γ₀(N).

Where Pith is reading between the lines

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

  • This explicit formula could be used to derive relations between periods or special values of L-functions for these Siegel forms.
  • Similar computations might be attempted for non-square-free levels by adjusting the test functions or local models.
  • The method may extend to other symplectic groups or higher genus Siegel forms.

Load-bearing premise

The square-free level condition is required for the global integral to factor in the stated way, along with the use of specific test functions in the Bessel models.

What would settle it

A calculation showing that the local zeta-integral for a particular test function and a square-free level does not match the claimed explicit evaluation would disprove the result.

read the original abstract

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels $\Gamma_0(N)$. On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.

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

0 major / 3 minor

Summary. The manuscript claims to explicitly compute the adelic Rankin-Selberg integral introduced by Piatetski-Shapiro for vector-valued Siegel cusp forms on GSp(2) of square-free level Γ₀(N), and to obtain exact evaluations of the associated local zeta integrals at specific test functions chosen in the Bessel models of irreducible admissible representations.

Significance. If the local computations are correct, the result supplies explicit formulas for these integrals in the square-free case, which is a concrete advance for integral representations of L-functions attached to Siegel modular forms; the square-free hypothesis is used only to guarantee the global integral factors into a product of locals, and the work supplies the missing local data rather than relying on abstract existence statements.

minor comments (3)
  1. [§2] §2, definition of the global integral: the precise normalization of the measure on the adelic quotient is not restated, which could affect the constant factors in the final formula; a one-sentence reminder would remove ambiguity.
  2. [§4] §4, local computations at finite places: the choice of the particular test function in the Bessel model is stated but the verification that it lies in the correct space is only sketched; adding one line of justification would make the argument self-contained.
  3. [Table 1] Table 1 (local factors): the entry for the unramified place uses a different normalization of the local L-factor than the one appearing in the global statement; a brief note reconciling the two would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The report correctly identifies that the manuscript supplies explicit local data for the Piatetski-Shapiro integral in the square-free case, which is the main contribution.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper claims an explicit computation of the adelic Rankin-Selberg integral for vector-valued Siegel cusp forms of square-free level, achieved by evaluating local zeta integrals at chosen test functions in the Bessel models. This is a direct local-global calculation under an explicit scope restriction (square-free level to ensure factorization), with no reduction of any claimed result to a fitted parameter, self-definition, or load-bearing self-citation. The derivation chain consists of standard local computations and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5573 in / 973 out tokens · 19359 ms · 2026-05-23T23:12:37.216127+00:00 · methodology

discussion (0)

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Reference graph

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