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arxiv: 2410.02625 · v4 · submitted 2024-10-03 · 🧮 math.NT · math.RT

The tower property on the genericity of global theta lifts

Jaeho Haan , Sanghoon Kwon This is my paper

Pith reviewed 2026-05-23 20:09 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords global theta liftsgenericitytower propertyBessel periodsFourier-Jacobi periodsL-functionsGan-Gross-Prasad conjectureclassical groups
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The pith

The first occurrence of global theta lifts between dual reductive groups preserves genericity.

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

The paper studies how Rallis' tower property extends to genericity for global theta lifts between classical groups. It proves that the first such lift between dual reductive groups preserves genericity by linking the analytic behavior of L-functions to the non-vanishing of special Bessel and Fourier-Jacobi periods. This preservation is then applied to confirm the global Gan-Gross-Prasad conjecture for SO_{2n+1} times SO_2 when the latter is split and carries the trivial representation. A reader would care because genericity controls many local and global properties of automorphic representations on these groups.

Core claim

By exploring the relationship between the analytic properties of L-functions and special Bessel and Fourier-Jacobi periods, we demonstrate that the first occurrence of global theta lifts between dual reductive groups preserves genericity. As an application, we establish the global Gan-Gross-Prasad conjecture for SO_{2n+1} × SO_2 under the assumption that SO_2 is split and its representation is trivial.

What carries the argument

The link between analytic properties of L-functions and non-vanishing of special Bessel and Fourier-Jacobi periods, which transfers genericity across the first theta lift in the tower.

If this is right

  • Genericity is preserved precisely at the first occurrence of the global theta lift between dual pairs.
  • The global Gan-Gross-Prasad conjecture holds for SO_{2n+1} × SO_2 when SO_2 is split and the representation is trivial.
  • The tower property for genericity applies across various classical groups via the same period-L-function relation.

Where Pith is reading between the lines

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

  • The same period analysis might apply to other dual pairs not treated in the paper, such as those involving unitary groups.
  • This global result could be combined with local theta correspondence to classify generic representations in the image of the lift without computing Fourier coefficients directly.

Load-bearing premise

The analytic properties of L-functions can be translated into statements about the vanishing or non-vanishing of special Bessel and Fourier-Jacobi periods to conclude that genericity is preserved.

What would settle it

An explicit pair of dual reductive groups and an automorphic representation where the first theta lift is non-generic, yet the associated L-function has a pole or zero that would force the periods to indicate genericity.

read the original abstract

In this paper, we examine the tower property concerning the genericity of global theta lifts between various classical groups, drawing inspiration from Rallis' tower property. By exploring the relationship between the analytic properties of $L$-functions and special Bessel and Fourier-Jacobi periods, we demonstrate that the first occurrence of global theta lifts between dual reductive groups preserves genericity. As an application, we establish the global Gan-Gross-Prasad conjecture for $\SO_{2n+1} \times \SO_{2}$ under the assumption that $\SO_{2}$ is split and its representation is trivial.

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 paper studies the tower property for genericity of global theta lifts between dual reductive groups, following Rallis' tower property. It establishes that the first occurrence of such a lift preserves genericity by relating analytic properties of L-functions to the non-vanishing of special Bessel and Fourier-Jacobi periods. As an application, the global Gan-Gross-Prasad conjecture is proved for SO(2n+1) × SO(2) when SO(2) is split and the representation on the second factor is trivial.

Significance. If the central claim holds, the result strengthens the Gan-Gross-Prasad program and the theory of theta correspondences by providing a genericity-preservation statement at the first occurrence of lifts. The approach via L-function analytic properties and period identities is consistent with existing methods in the field and offers a natural extension of Rallis' ideas to genericity questions.

minor comments (3)
  1. [Abstract] The abstract states the main result in terms of 'dual reductive groups' but does not specify the precise pairs of groups for which the tower property is proved; adding a sentence listing the classical groups under consideration would improve clarity.
  2. [Application section (likely §5 or §6)] In the application to the GGP conjecture, the assumption that the representation on SO(2) is trivial is used to reduce the period; a brief remark on whether the argument adapts to non-trivial characters would help readers assess the scope.
  3. [Preliminaries] Notation for the special Bessel and Fourier-Jacobi periods should be introduced with explicit references to the local and global definitions used in the period identities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work on the tower property for genericity of global theta lifts and its application to the Gan-Gross-Prasad conjecture. The recommendation of minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external analytic relationships

full rationale

The paper's central claim—that the first occurrence of global theta lifts preserves genericity—is derived from the relationship between L-function analytic properties and non-vanishing of Bessel/Fourier-Jacobi periods, drawing on Rallis' tower property and the Gan-Gross-Prasad framework. No equations or steps reduce by construction to author-defined inputs, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract and description present this as a consequence of standard methods in theta correspondence without internal redefinition or smuggling of ansatzes. The SO(2n+1) × SO(2) application is a direct specialization. This is the expected non-finding for a paper whose logic chain remains externally grounded.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions from the theory of automorphic L-functions and theta correspondence; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Analytic properties of L-functions control the non-vanishing of Bessel and Fourier-Jacobi periods
    Invoked to link L-function behavior to genericity preservation (abstract)

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