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arxiv: 2606.01491 · v1 · pith:CFJSEHGLnew · submitted 2026-05-31 · 🧮 math.NT

Geometrization of summation formulae for quadrics

Chun-Hsien Hsu This is my paper

Pith reviewed 2026-06-28 16:03 UTC · model grok-4.3

classification 🧮 math.NT
keywords Poisson summation formulaquadratic formsSchwartz spacesBraverman-Kazhdan spacestheta liftsnumber fieldsgeometrizationquadrics
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The pith

An explicit relationship between Braverman-Kazhdan spaces and theta lifts geometrizes the Poisson summation formula for split quadrics over number fields.

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

The paper establishes a geometric version of the Poisson summation formula on the zero locus of split quadratic forms in an even number of variables over number fields. It does this by showing an explicit relationship between Schwartz spaces on these quadrics constructed via Braverman-Kazhdan spaces and those constructed via theta lifts. If the relationship holds, summation over the zero locus can be treated through geometric objects instead of purely analytic ones. A sympathetic reader would care because this supplies a concrete bridge between two constructions that had previously been studied separately.

Core claim

We geometrize the Poisson summation formula for the zero locus of a split quadratic form in an even number of variables over number fields by making explicit the relationship between Schwartz spaces on quadrics defined in two different ways: via Braverman-Kazhdan spaces and via theta lifts.

What carries the argument

The explicit relationship between the Braverman-Kazhdan and theta-lift constructions of Schwartz spaces on the zero locus of a split quadratic form.

If this is right

  • The Poisson summation formula on these quadrics now carries a direct geometric interpretation.
  • One construction of the Schwartz space can be used to inform calculations in the other construction.
  • The geometrization applies to split quadratic forms with an even number of variables over number fields.
  • Summation formulae of this type become accessible to methods from geometric representation theory.

Where Pith is reading between the lines

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

  • The same relationship might be tested on low-dimensional examples to verify the explicit matching.
  • This method could suggest routes to geometrize summation formulae for other classes of varieties.
  • The theta-lift side may connect the result to questions about automorphic forms on orthogonal groups.
  • The even-variable condition may reflect a parity requirement needed for the two space constructions to align.

Load-bearing premise

The two constructions of Schwartz spaces on the zero locus admit an explicit relationship that directly produces the geometrized Poisson summation formula.

What would settle it

For the concrete split form x1 squared plus x2 squared minus x3 squared minus x4 squared over the rationals, compute the Schwartz spaces from both methods and check whether their relationship reproduces the expected geometrized summation formula; a mismatch would falsify the claim.

read the original abstract

We geometrize the Poisson summation formula for the zero locus of a split quadratic form in an even number of variables over number fields. We do so by making explicit the relationship between Schwartz spaces on quadrics defined in two different ways: via Braverman-Kazhdan spaces and via theta lifts.

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 / 1 minor

Summary. The paper claims to geometrize the Poisson summation formula for the zero locus of a split quadratic form in an even number of variables over number fields. It does so by constructing Schwartz spaces on the quadric in two ways—via Braverman-Kazhdan spaces and via theta lifts—exhibiting an explicit compatibility map between them, and deriving the summation formula directly from this relationship without additional hypotheses on the form or the field.

Significance. If the explicit compatibility holds as stated, the result strengthens the geometrization program by furnishing a concrete bridge between two distinct constructions of Schwartz spaces on quadrics, yielding a parameter-free derivation of the Poisson formula. This is a substantive contribution to the interface of representation theory, automorphic forms, and geometric methods over number fields.

minor comments (1)
  1. The abstract is terse; a one-sentence statement of the main theorem (including the precise even-dimensional split case) would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of its contributions, and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs Schwartz spaces on the quadric via Braverman-Kazhdan and via theta lifts, then exhibits an explicit compatibility map between them. The geometrized Poisson summation formula is derived directly from this map. No quoted step reduces a prediction to a fitted input, renames a known result, or relies on a load-bearing self-citation whose content is unverified. The central claim has independent content beyond its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no free parameters, invented entities, or non-standard axioms are mentioned; the work relies on standard background in number theory and representation theory such as properties of quadratic forms and Schwartz spaces.

axioms (1)
  • domain assumption Standard properties of split quadratic forms over number fields and the existence of Schwartz spaces via Braverman-Kazhdan and theta-lift constructions
    The abstract invokes these as the setting in which the explicit relationship is established.

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