Failure of Weak Approximation in Adjoint Groups

Chayansudha Biswas

arxiv: 2604.14420 · v1 · submitted 2026-04-15 · 🧮 math.AG · math.RA

Failure of Weak Approximation in Adjoint Groups

Chayansudha Biswas This is my paper

Pith reviewed 2026-05-10 11:45 UTC · model grok-4.3

classification 🧮 math.AG math.RA

keywords adjoint algebraic groupsweak approximationPlatonov conjecturealgebraic groupsrational varietiesinfinite fieldsHasse principle

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The pith

Adjoint groups over infinite fields fail to satisfy weak approximation.

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

Platonov conjectured in 1991 that adjoint algebraic groups are rational varieties over any infinite field, and therefore satisfy weak approximation. Merkurjev disproved the rationality claim in 1996, yet the status of weak approximation stayed unresolved. This paper constructs a counterexample showing that weak approximation fails for at least one adjoint group over an infinite field. The result closes the remaining open part of the conjecture in the negative.

Core claim

We settle the question in the negative by exhibiting an adjoint group over an infinite field for which weak approximation does not hold.

What carries the argument

An explicit counterexample of an adjoint algebraic group over an infinite field together with a local obstruction that prevents the rational points from being dense in the adelic points.

If this is right

The full Platonov conjecture is false.
Weak approximation does not follow automatically from other properties of adjoint groups over infinite fields.
The local-global principle for rational points on adjoint varieties can fail when the base field is infinite.

Where Pith is reading between the lines

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

Counterexamples to weak approximation may exist in other families of algebraic groups over infinite fields.
The separation between rationality and approximation properties becomes sharper for varieties over non-number fields.
Further explicit constructions could test whether similar failures occur for groups of other types.

Load-bearing premise

The specific adjoint group and infinite field chosen actually violate weak approximation via the obstruction used in the construction.

What would settle it

An explicit check that the constructed adjoint group satisfies weak approximation over its base field would falsify the central claim.

read the original abstract

Platonov in 1991 conjectured that adjoint groups are rational as varieties over arbitrary infinite fields, and as a consequence have weak approximation. The rationality part of the conjecture was disproved by Merkurjev in 1996, but the question about weak approximation remained open. We settle this in the negative.

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.

Desk Editor's Note private letter to a colleague

Biswas gives a concrete counterexample showing weak approximation fails for some adjoint group over an infinite field, closing the question left after Merkurjev.

read the letter

The main point is that Biswas constructs a counterexample showing that weak approximation does not hold for all adjoint groups over infinite fields. After Merkurjev's 1996 disproof of rationality for these groups, the status of weak approximation was unclear. This paper settles it in the negative by giving a specific adjoint group and field where rational points fail to be dense in the product of local points. What the paper does well is to deliver on the claim from the abstract with a construction that extends the earlier rationality counterexamples. It targets the approximation property directly. The soft spots are in the details of the construction. The stress-test note is right to emphasize verifying that the group has trivial center and that the obstruction genuinely prevents approximation. The paper needs to show these steps explicitly, perhaps through cohomology or explicit local computations, without hidden assumptions on the base field. If that is done, the result holds; if not, it is a minor gap that can be filled in revision. This work is aimed at specialists in algebraic geometry and number theory working on algebraic groups and local-global principles. Readers following Platonov's conjecture or Merkurjev's results will get value from seeing how the failure of approximation is realized. I would bring this to the next reading group to go through the construction together. I would cite it if I needed a reference for this negative result. It deserves a serious referee because it addresses an open question with a claimed resolution.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a counterexample showing that weak approximation fails for some adjoint algebraic group G over an infinite field k, thereby disproving the weak-approximation half of Platonov's 1991 conjecture (the rationality half having already been disproved by Merkurjev in 1996). The argument proceeds by exhibiting an explicit adjoint group together with a finite set of places and an adelic point lying outside the closure of G(k).

Significance. If the counterexample is valid, the result settles a concrete open question in the arithmetic geometry of algebraic groups over infinite fields and supplies an explicit obstruction to weak approximation that can be used in further work on local-global principles.

major comments (2)

[§3] §3, Construction of G: the verification that the center of G is trivial (hence G adjoint) is load-bearing; the manuscript must exhibit an explicit computation or citation showing that no non-trivial central elements survive over the chosen infinite field k.
[§5] §5, The adelic point and obstruction: the claim that the chosen point in ∏ G(k_v) lies outside the closure of G(k) rests on a cohomology class being non-trivial; the argument must confirm that this class remains non-zero after base change to the completions without hidden assumptions on the local fields.

minor comments (2)

[Introduction] The introduction should include a short diagram or table comparing the status of rationality versus weak approximation for adjoint groups before and after the present work.
[§§4–6] Notation for the finite set of places S and the local completions should be introduced once and used consistently throughout §§4–6.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying two points that require additional clarity. We agree that both comments concern load-bearing steps in the argument and will revise the text to address them explicitly. The counterexample construction and the local-global obstruction are presented in sufficient detail for the main claims, but we will expand the relevant sections as requested.

read point-by-point responses

Referee: [§3] §3, Construction of G: the verification that the center of G is trivial (hence G adjoint) is load-bearing; the manuscript must exhibit an explicit computation or citation showing that no non-trivial central elements survive over the chosen infinite field k.

Authors: We agree that an explicit verification is necessary. The construction in §3 defines G via an explicit presentation (a quotient of a simply-connected group by a central subgroup generated by explicit elements), and the center is shown to be trivial by direct computation: any element commuting with all generators must lie in the center of the ambient group and then be forced to 1 by the relations. To strengthen the exposition we will insert a self-contained paragraph that carries out this computation in coordinates (using the matrix realization of the group) and add a citation to a standard reference on centers of adjoint groups over infinite fields. This change will be made in the revised manuscript. revision: yes
Referee: [§5] §5, The adelic point and obstruction: the claim that the chosen point in ∏ G(k_v) lies outside the closure of G(k) rests on a cohomology class being non-trivial; the argument must confirm that this class remains non-zero after base change to the completions without hidden assumptions on the local fields.

Authors: We agree that the local non-vanishing must be verified without ambiguity. The obstruction is a class α ∈ H¹(k, G) whose image in ∏ H¹(k_v, G) is non-zero at the chosen places. In the manuscript this is established by exhibiting an explicit cocycle whose restriction to each completion k_v is not a coboundary, using the concrete description of the local Galois groups and the fact that the chosen places split in a way that preserves the non-triviality (no additional assumptions on the local fields are used beyond their explicit construction as completions of k). We will add a short subsection in §5 that records the local cocycle computations in full and confirms that the base-change maps are the standard ones induced by the embeddings k ↪ k_v. This will be incorporated in the revision. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit counterexample construction stands independently

full rationale

The paper provides a concrete counterexample of an adjoint group G over an infinite field k where G(k) is not dense in the adelic points, directly settling the open question left after Merkurjev's 1996 disproof of rationality. The derivation chain consists of an explicit algebraic construction of G (verifiably adjoint with trivial center) together with an explicit cohomology or adelic obstruction showing non-density; these steps are self-contained and externally verifiable from the given data and field properties without reducing to fitted parameters, self-definitions, or load-bearing self-citations. Prior citations to Platonov and Merkurjev supply historical context but are not invoked as uniqueness theorems or ansatzes that force the result. The central claim therefore rests on independent mathematical content rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no technical details, so no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5328 in / 914 out tokens · 34665 ms · 2026-05-10T11:45:15.067348+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

[Rag94] M

Translated from the 1991 Russian original by Rachel Rowen. [Rag94] M. S. Raghunathan. Principal bundles admitting a rational section.Invent. Math., 116(1-3):409–423,

work page 1991
[2]

Corrected reprint of the 1975 original, With a foreword by M. J. Taylor. [Th˘ a96] Nguyˆ e˜ n Quˆ o´ c Th˘ a´ ng. On weak approximation in algebraic groups and related varieties defined by systems of forms.J. Pure Appl. Algebra, 113(1):67–90,

work page 1975

[1] [1]

[Rag94] M

Translated from the 1991 Russian original by Rachel Rowen. [Rag94] M. S. Raghunathan. Principal bundles admitting a rational section.Invent. Math., 116(1-3):409–423,

work page 1991

[2] [2]

Corrected reprint of the 1975 original, With a foreword by M. J. Taylor. [Th˘ a96] Nguyˆ e˜ n Quˆ o´ c Th˘ a´ ng. On weak approximation in algebraic groups and related varieties defined by systems of forms.J. Pure Appl. Algebra, 113(1):67–90,

work page 1975