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arxiv: 2604.03095 · v1 · submitted 2026-04-03 · 🧮 math.NT · math.RT

Functoriality and the theta correspondence

Alexander Hazeltine This is my paper

Pith reviewed 2026-05-13 19:13 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords theta correspondencefunctorialityp-adic groupsABV-packetsAdams conjectureclassical groupslocal representationsLanglands correspondence
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The pith

The functoriality of the local theta correspondence for classical p-adic groups is realized by adapting the Adams conjecture to ABV-packets.

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

The paper investigates how the local theta correspondence between representations of classical groups over p-adic fields preserves functoriality. It shows that this behavior can be captured by modifying the Adams conjecture to apply to ABV-packets instead of its usual setting. Evidence supporting the adapted conjecture is supplied, with the strongest support appearing when the groups involved are general linear. A sympathetic reader would care because this link would tie the theta correspondence more tightly to the local Langlands correspondence and packet structures.

Core claim

The functoriality of the local theta correspondence for classical p-adic groups is realized via the adaptation of the Adams conjecture to ABV-packets. Evidence for the adapted conjecture is provided, especially in the case of general linear groups.

What carries the argument

The adaptation of the Adams conjecture to ABV-packets, which carries the argument by encoding how the local theta correspondence respects functorial transfers between groups.

If this is right

  • Functoriality of the local theta correspondence holds for classical p-adic groups once the adapted conjecture is assumed.
  • The theta correspondence maps ABV-packets to ABV-packets in a manner compatible with the Adams conjecture.
  • The approach yields verifiable statements for representations of general linear groups over p-adic fields.
  • The local theta lift respects the packet structure used in the local Langlands correspondence.

Where Pith is reading between the lines

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

  • The same adaptation might supply a uniform description of theta lifts across all classical groups rather than case-by-case verification.
  • If the conjecture holds, it could be used to test or refine other conjectures linking local packets to global automorphic forms.
  • The method suggests a way to compute explicit theta correspondences for low-rank classical groups by direct comparison with known Adams-conjecture data.

Load-bearing premise

The adapted Adams conjecture to ABV-packets correctly encodes the functoriality properties of the local theta correspondence.

What would settle it

A concrete counterexample in which the theta lift of a representation fails to match the prediction given by the adapted Adams conjecture for some irreducible representation of a classical p-adic group.

read the original abstract

We study the functoriality of the local theta correspondence for classical $p$-adic groups. This is realized via the adaptation of the Adams conjecture to ABV-packets. We provide evidence for the conjecture, especially in the case of general linear groups.

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 paper claims that the functoriality of the local theta correspondence for classical p-adic groups is realized by adapting the Adams conjecture to ABV-packets, with explicit supporting calculations provided especially in the case of general linear groups.

Significance. If the proposed adaptation correctly encodes the packet-to-packet maps and L-parameter matching for all classical dual pairs (including Sp/O and U/U), the work would supply a concrete mechanism linking the theta correspondence to functoriality in the local Langlands correspondence, building on existing evidence for GL groups. The GL calculations constitute a verifiable strength; extension to the remaining groups would be a substantial contribution if the intertwining operators are shown to match without ad-hoc corrections.

major comments (1)
  1. [Abstract] Abstract and central claim: the assertion that the adaptation of the Adams conjecture to ABV-packets realizes functoriality for all classical p-adic groups rests on the unverified statement that the same construction preserves the required intertwining operators and L-parameter matching for Sp/O and U/U dual pairs; only GL evidence is supplied, so the general claim is not yet load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and valuable feedback on our manuscript. We address the major comment below and will revise the paper accordingly to clarify the scope of our claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract and central claim: the assertion that the adaptation of the Adams conjecture to ABV-packets realizes functoriality for all classical p-adic groups rests on the unverified statement that the same construction preserves the required intertwining operators and L-parameter matching for Sp/O and U/U dual pairs; only GL evidence is supplied, so the general claim is not yet load-bearing.

    Authors: We agree that the explicit verification of the intertwining operators and L-parameter matching is provided in detail only for general linear groups, while the extension to Sp/O and U/U dual pairs relies on the adapted Adams conjecture holding uniformly without additional corrections. The manuscript presents the adaptation as a proposed mechanism realizing functoriality across classical p-adic groups, supported by the GL calculations as the primary evidence. To address this, we will revise the abstract to emphasize that the construction realizes the functoriality under the adapted conjecture, with concrete evidence supplied especially for GL groups. We will also add a clarifying paragraph in the introduction discussing the conjectural status for the remaining dual pairs and the absence of ad-hoc adjustments in the framework. This revision will make the central claim more precise while preserving the manuscript's core contribution. revision: yes

Circularity Check

0 steps flagged

No circularity detected; central claim framed as conjecture with external evidence for GL case

full rationale

The abstract and provided excerpts present the functoriality of the local theta correspondence as realized by adapting the Adams conjecture to ABV-packets, supplying evidence especially for general linear groups. No equations, definitions, or self-citations are quoted that reduce the claimed realization to a fitted input, self-definition, or load-bearing prior result by the same authors. The derivation is therefore treated as self-contained against the external Adams conjecture benchmark, with the partial evidence for GL groups providing independent content rather than a renaming or construction that forces the outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the claim rests on the unstated assumption that ABV-packets behave compatibly with the Adams conjecture.

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