On the rationality of the Weil Representation and the local theta correspondence
Pith reviewed 2026-05-16 11:51 UTC · model grok-4.3
The pith
The Weil representation over non-archimedean local fields descends to a number field, making the local theta correspondence rational.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The Weil representation over a non-archimedean local field can be realised with coefficients in a number field via an explicit descent argument, and the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field; these two results together show that the classical local theta correspondence is rational.
What carries the argument
Explicit descent of the Weil representation to number fields and more general coefficient rings and fields, which transfers the validity of the theta correspondence from algebraic closures to the base perfect fields.
If this is right
- The local theta correspondence can be defined with coefficients in number fields rather than requiring algebraic closures.
- The correspondence extends directly to ℓ-modular coefficients and to families over rings of integers.
- Validity checks for the correspondence over perfect fields reduce to checks over algebraic closures.
Where Pith is reading between the lines
- The descent in families suggests the possibility of deforming representations while preserving the theta correspondence.
- Rationality may allow direct comparison between local theta lifts and global arithmetic constructions without base change.
Load-bearing premise
The technical properties of the Weil representation allow it to descend over the stated coefficient rings and fields in the way the explicit argument requires.
What would settle it
An explicit computation for a concrete non-archimedean local field and a concrete coefficient ring showing that the Weil representation fails to descend to any number field, or a pair of perfect field and its algebraic closure where the theta correspondence holds on one but not the other.
read the original abstract
We prove that the Weil representation over a non-archimedean local field can be realised with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell$-modular coefficient fields, as well as coefficient rings such as rings of integers i.e. in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that the Weil representation over a non-archimedean local field can be realized with coefficients in a number field, giving an explicit descent to the cyclotomic extension generated by the values of a fixed nontrivial additive character ψ on the residue field. The 2-cocycle is shown to take values in this field by direct computation of the Weil and Maslov indices on a symplectic basis, with technical properties verified over the stated coefficient rings by reduction to finite fields via Hensel lifting. It further proves that the local theta correspondence over a perfect field k is valid if and only if it is valid over the algebraic closure of k (Theorem 6.2), via Galois descent on the matrix coefficients of the Weil representation. These results together establish the rationality of the classical local theta correspondence.
Significance. If the results hold, this provides a foundational rationality statement for the Weil representation and local theta correspondence, allowing descent to number fields, ℓ-modular coefficients, and families over rings of integers. The explicit construction in §§3–5 and the Galois descent argument in Theorem 6.2 are strengths that enable arithmetic applications and studies in families.
major comments (1)
- [§§3–5] §§3–5: The descent of the 2-cocycle via Hensel lifting from the finite-field case is central to the rationality claim. Please clarify how the cocycle identity on generators is preserved under the lifting, including any conditions on the ramification or the choice of symplectic basis.
minor comments (2)
- [Theorem 6.2] Theorem 6.2: The term 'valid' for the theta correspondence should be recalled briefly in the statement or immediately preceding paragraph for clarity.
- [Abstract] Abstract: The claim of applicability to ℓ-modular coefficient fields and rings of integers is stated but would benefit from an explicit pointer to the section(s) containing the relevant arguments.
Simulated Author's Rebuttal
We are grateful to the referee for their positive assessment and for highlighting the need for additional clarification on the Hensel lifting argument in §§3–5. We will revise the manuscript to address this point explicitly.
read point-by-point responses
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Referee: [§§3–5] §§3–5: The descent of the 2-cocycle via Hensel lifting from the finite-field case is central to the rationality claim. Please clarify how the cocycle identity on generators is preserved under the lifting, including any conditions on the ramification or the choice of symplectic basis.
Authors: We appreciate this comment. The cocycle identity is first verified directly over finite fields by explicit computation of the Weil and Maslov indices with respect to a fixed symplectic basis (the standard hyperbolic basis). This verification consists of checking a finite number of polynomial identities in the matrix entries. Since these identities hold over the residue field, Hensel's lemma applies directly to lift the solutions to the p-adic integers, as the defining equations are integral and the derivative conditions are satisfied due to the non-degeneracy of the symplectic form. No additional ramification conditions are required beyond the field being non-archimedean local with residue field of characteristic not 2 (as assumed throughout). The symplectic basis is chosen to be the standard one, which lifts canonically from the finite field case without ramification issues. We will add a detailed remark in §4 clarifying this preservation mechanism. revision: yes
Circularity Check
No significant circularity; derivation is constructive and self-contained
full rationale
The paper constructs the Weil representation explicitly over a cyclotomic extension generated by values of a fixed additive character, computing the 2-cocycle directly via Weil and Maslov indices on a symplectic basis, then verifies equivariance and compatibility over coefficient rings by Hensel lifting to finite fields and checking the cocycle identity on generators. The theta correspondence equivalence (Theorem 6.2) follows from Galois descent: if a representation descends to the perfect field k then its theta lift descends because matrix coefficients of the Weil representation are defined over k. These steps are independent constructive arguments using standard tools (Hensel lifting, Galois descent) and do not reduce the rationality claim to any fitted input, self-definition, or self-citation chain by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard properties of the Weil representation and theta correspondence over local fields
discussion (0)
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