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arxiv: 2604.18436 · v1 · submitted 2026-04-20 · 🧮 math.AG · math.NT

Unirational algebraic groups and tame ramification

Otto Overkamp , Ismaele Vanni This is my paper

Pith reviewed 2026-05-10 03:21 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords unirational algebraic groupsNéron lft-modelsEdixhoven filtrationmotivic zeta functionstame ramificationalgebraic toriabelian varietiesmultiplicative reduction
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The pith

Unirational algebraic groups have rational jumps in their Néron models and rational motivic zeta functions.

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

The paper examines how the Néron lft-model of a smooth connected commutative algebraic group G without a copy of Ga changes when base-changed to ramified extensions K(d) for d prime to the residue characteristic. This behaviour is captured by the jumps of Edixhoven's filtration on the special fibre and by Halle-Nicaise's motivic zeta function. When G is unirational, for example an algebraic torus, these jumps are rational numbers and the zeta function is a rational function. The same rationality conclusions are deduced for abelian varieties with potentially totally multiplicative reduction. Along the way the work settles questions posed by Halle-Nicaise, Edixhoven and Oesterlé.

Core claim

If G is unirational, the jumps of G are rational numbers and the motivic zeta function of G is a rational function. We also deduce analogous results for Abelian varieties with potentially totally multiplicative reduction. This answers a question of Halle-Nicaise and partially one of Edixhoven. Along the way, we answer a question of Oesterlé about the structure of unipotent algebraic groups over function fields in positive characteristic. Under stronger conditions on G, we obtain rationality of jumps even for separably closed but imperfect k.

What carries the argument

The jumps in Edixhoven's filtration on the special fibre of the Néron lft-model, which record the change under base change to K(d), together with Halle-Nicaise's motivic zeta function.

If this is right

  • Jumps of Edixhoven's filtration are rational numbers for any unirational G.
  • The motivic zeta function of a unirational G is a rational function.
  • Abelian varieties with potentially totally multiplicative reduction have rational jumps and rational motivic zeta functions.
  • Rationality of jumps continues to hold when the residue field is separably closed but imperfect, provided stronger conditions on G.
  • A question of Oesterlé on the structure of unipotent groups over function fields in positive characteristic is answered.

Where Pith is reading between the lines

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

  • The rationality statements may allow concrete computation of motivic zeta functions for algebraic tori over local fields.
  • The clarification of unipotent groups over function fields could feed into broader classification questions for algebraic groups in positive characteristic.
  • The base-change analysis under tame ramification might be adapted to other classes of groups once a suitable replacement for unirationality is identified.

Load-bearing premise

The group G must be unirational in addition to being smooth, connected, commutative and without a copy of Ga.

What would settle it

An explicit unirational G whose motivic zeta function fails to be rational or whose jumps include an irrational number.

read the original abstract

Let $\mathcal{O}_K$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k.$ Let $G$ be a smooth connected commutative algebraic group over $K$ which does not contain a copy of $\mathbf{G}_{\mathrm{a}}.$ For each $d$ prime to $p:=\mathrm{char}\, k,$ let $K(d)$ be the unique extension of $K$ of degree $d.$ We investigate how the N\'eron lft-model of $G$ behaves under base change to the ring of integers $\mathcal{O}_{K(d)}.$ Information about this behaviour is encoded in the "jumps" of Edixhoven's filtration on the special fibre of the N\'eron lft-model of $G,$ as well as in Halle-Nicaise's motivic zeta function of $G.$ If $G$ is unirational (e. g. an algebraic torus), we show that the jumps of $G$ are rational numbers and that the motivic zeta function of $G$ is a rational function. We also deduce analogous results for Abelian varieties with potentially totally multiplicative reduction. This answers a question of Halle-Nicaise and partially one of Edixhoven. Along the way, we answer a question of Oesterl\'e about the structure of unipotent algebraic groups over function fields in positive characteristic. Under stronger conditions on $G,$ we obtain rationality of jumps even for separably closed but imperfect $k.$

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 Néron lft-model of a smooth connected commutative algebraic group G over a complete DVR K with algebraically closed residue field k (no Ga factor), focusing on its behavior under tame base change to K(d) for d prime to p. It encodes this via jumps in Edixhoven's filtration on the special fiber and Halle-Nicaise's motivic zeta function. Under the additional hypothesis that G is unirational, the jumps are shown to be rational and the motivic zeta function rational; analogous results are deduced for abelian varieties with potentially totally multiplicative reduction. The work resolves questions of Halle-Nicaise and Edixhoven, and as a byproduct answers Oesterlé's question on unipotent groups over function fields in positive characteristic. Rationality of jumps is also obtained under stronger hypotheses when k is separably closed but imperfect.

Significance. If the proofs hold, the results establish rationality of key arithmetic invariants (jumps and motivic zeta functions) for unirational groups, which is a substantive advance in the study of tame ramification and Néron models. The resolutions of the cited open questions, together with the deduction for abelian varieties, give the work clear significance in arithmetic geometry. The byproduct resolution of Oesterlé's question on unipotent groups is an additional strength.

minor comments (3)
  1. The introduction would benefit from a short paragraph recalling the precise definition of Edixhoven's filtration and the jumps (currently referenced only by name in the abstract and §1), to make the main theorems more immediately accessible.
  2. Notation for the motivic zeta function Z_G(t) is introduced without an explicit formula or reference to Halle-Nicaise's original definition in the first section where it appears; adding this would improve readability.
  3. In the statement of the main theorem on rationality of jumps (likely Theorem 1.1 or 3.1), the hypothesis that G contains no copy of Ga is stated but its necessity is not illustrated by a brief counter-example or remark; a short sentence would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity; results are conditional theorems under explicit unirationality hypothesis

full rationale

The paper states its main claims as theorems conditioned on G being unirational (smooth, connected, commutative, no Ga factor) over a complete DVR with algebraically closed residue field. Jumps are shown rational and the motivic zeta function rational under this hypothesis; analogous results for abelian varieties follow from an additional reduction assumption. No equations or steps reduce by construction to fitted inputs, self-definitions, or self-citation chains. External questions (Halle-Nicaise, Edixhoven, Oesterlé) are answered as byproducts rather than used as load-bearing premises. The derivation is self-contained against standard algebraic geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard background in algebraic groups, Néron models, and filtrations; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption G is a smooth connected commutative algebraic group over K without a copy of Ga
    Core setup assumption defining the objects whose Néron models are studied.
  • domain assumption k is algebraically closed and d is prime to p = char k
    Ensures unique unramified extensions K(d) and controls the ramification behavior.

pith-pipeline@v0.9.0 · 5572 in / 1345 out tokens · 38361 ms · 2026-05-10T03:21:25.267044+00:00 · methodology

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