Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$

Abdulmuhsin Alfaraj

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Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$

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Pith's one-line read For smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over global function fields, the paper proves the meromorphic continuation and pole locations of the height zeta function, and identifies the leading constant with the…

desk verdict The main theorem rests on an unjustified step in Proposition 8.3(ii): the logarithmic derivative map on F_v^* is not surjective in characteristic p, so the reduction to d_α coprime to p fails and the local Fourier transform bound is unproven. read the letter →

arxiv 2505.04562 v1 pith:ZKZFPNSX submitted 2025-05-07 math.NT math.AG

Abdulmuhsin Alfaraj This is my paper · ORCID

classification math.NTmath.AG MSC 11G5014G0514G1011M38

keywords heightzetafunctionrationalpointsofboundedequivariantcompactificationsformstheadditivegroupglobalfieldsF-woundgroupsPoissonsummationformulaTamagawameasures

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes the function-field analogue of the rational-point distribution conjecture for smooth equivariant compactifications of forms of the additive group $\mathbb{G}_a^n$, conditional on a hypothesis about the inseparable part of the boundary. Over a global function field, such a form can be a nontrivial $p$-torsion twist, so the boundary of a smooth compactification may have components that are not geometrically reduced; the paper shows what must be assumed about these components for the height zeta function to have the expected analytic shape. Under those assumptions, the height zeta function has a meromorphic continuation whose poles of maximal order lie on an arithmetic progression, and the leading constant equals the Tamagawa-type constant predicted by the conjecture. A second theorem proves that any commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation, which is the key step in matching the leading constant. A detailed example, the projective space $\mathbb{P}^{p-1}$ as a compactification of an $F$-wound group, illustrates the new inseparable phenomena and verifies the hypotheses.

What carries the argument

The central object is the height zeta function $Z_\lambda(s)$, and the engine of the proof is Poisson summation on the locally compact group $G(\mathbb{A}_F)$. The height is $K$-invariant for a compact open subgroup $K$, so the zeta function becomes a finite sum of Fourier transforms $\widehat H(\Psi;s\lambda)$ over characters of the finite quotient $G(\mathbb{A}_F)/(G(F)+K)$. At the trivial character, the local Fourier transform is computed by an explicit local formula at good places: the integral over the residue class of a point $\bar x$ is a product over the boundary components through $\bar x$, with factors $(q_v-1)/(q_v^{1+s_\alpha-\rho_\alpha}-1)$ for smooth components and $q_v^{\beta(\bar x)(\rho_\beta-s_\beta)}$ for inseparable components. Nontrivial characters are interpreted via an identification of characters of $G(\mathbb{A}_F)$ with elements of the Brauer group of $G$, built from additive characters and cup products in the Brauer group; this lets the author replace a pole order divisible by $p$ by a prime-to-$p$ order and thereby control the local Fourier transforms.

What would settle it

Take any smooth equivariant compactification satisfying Assumption 1.2 for which Condition 9.1 can be checked to fail (for instance, a form with Pic(G) nonzero and a character whose divisor has d_β>0 on an inseparable component), and compute the local Fourier transform at a good place; the theorem predicts no pole of maximal order b_λ outside the set s_j, so exhibiting such a pole in the height zeta function would refute the claimed sufficiency of the hypotheses.

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Extended reading notes

Core claim

Let $X$ be a smooth equivariant compactification of an $F$-form $G$ of $\mathbb{G}_a^n$ over a global function field, with boundary divisor $D=\sum_{\alpha\in A}D_\alpha$, and let $B\subset A$ index the components that are not geometrically reduced. Assuming that the reduced part of $D$ has strict normal crossings, that each inseparable component is geometrically irreducible with no $F_v$-points, has a $q$-metric taking finitely many values, and satisfies a finite-field point-counting estimate, plus Condition 9.1 for all nontrivial characters, Theorem 1.4 proves that for every big line bundle $L_\lambda$ the height zeta function $Z_\lambda(s)=\sum_{x\in G(F)}H_\lambda(x)^{-s}$ converges absolutely for $\Re(s)>a_\lambda$, continues meromorphically to $\Re(s)>a_\lambda-\delta$, and has poles of maximal order $b_\lambda$ exactly at $s_j=a_\lambda+j(2\pi i)/(d_\lambda\log q)$ for $j\in J_\lambda$, with nonzero leading constant $c_\lambda$. For the anticanonical height, Theorem 1.8 identifies $c_\rho$ as $\alpha^*(X)\,\tau_X(X(\mathbb{A}_F))$, the product of the effective-cone constant and the Tamagawa measure of adelic points, matching the standard prediction. The proof also shows that such $G$ have trivial Tate-Shafarevich group and satisfy weak approximation, so the Tamagawa number equals $|\mathrm{Pic}(G)|$ and the Brauer group of $X$ collapses to that of $F$.

Load-bearing premise

Everything rests on Condition 9.1, that every non-trivial character appearing in the Poisson sum can be represented by a function whose divisor has no poles along the non-geometrically reduced boundary components; the paper verifies this only in special cases and gives no general criterion for it.

Editorial extensions

If this is right

For split $\mathbb{G}_a^n$ over a global function field, the theorem gives the full rational-point distribution conjecture for smooth equivariant compactifications with strict normal crossings boundary, with the predicted leading constant.
For any admissible $L_\lambda$ with integral coordinates, the weighted average of point counts satisfies an asymptotic power law with exponent $a_\lambda$ and growth order $M^{b_\lambda-1}$; when $d_\lambda\mid g_\lambda$, the individual counts vanish unless $d_\lambda\mid M$ and then follow the same asymptotic.
The leading constant for the anticanonical height is $\alpha^*(X)\tau_X(X(\mathbb{A}_F))$, so the Tamagawa measure, not an ad hoc normalization, gives the correct constant.
Every connected commutative unipotent group admitting a smooth equivariant compactification has trivial Tate-Shafarevich group and satisfies weak approximation; consequently its Tamagawa number equals $|\mathrm{Pic}(G)|$.
The method covers concrete new cases such as $\mathbb{P}^{p-1}$ compactifying $\operatorname{Res}_{F^{1/p}/F}\mathbb{G}_m/\mathbb{G}_m$, where the count is explicitly $N(\omega_X^{-1},M)\sim \tfrac{1}{p}\bigl(q^{p-1}\operatorname{Res}_{s=1}\zeta_F(s)\prod_v C_v\bigr)\log(q)\,q^M$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step, not taken in the paper, is to prove Condition 9.1 for all forms admitting a smooth equivariant compactification; the Brauer-group description of characters suggests such a proof would reduce to showing that inseparable boundary components never carry the divisor of a character contributing to the pole of maximal order.
The $\mathbb{P}^{p-1}$ construction is the first member of a family $\mathbb{P}^{p^k-1}$ compactifying $\operatorname{Res}_{F^{1/p^k}/F}\mathbb{G}_m/\mathbb{G}_m$, so the same analysis should yield explicit asymptotics for those cases and for products with split $\mathbb{G}_a^n$-factors, exactly as the paper indicates in its remarks.
The Hasse-principle theorem may hold for a broader class than the zeta-function theorem: any commutative unipotent group with a smooth equivariant compactification, whether or not the boundary metric conditions hold, would have trivial Brauer-Manin obstruction to weak approximation, so the arithmetic of these groups is controlled by their Picard groups alone.
If Condition 9.1 were replaced by a weaker cancellation mechanism between the Fourier transforms of the inseparable components, the main theorem would extend to all smooth equivariant compactifications; the local estimates indicate the obstruction is concentrated in the contribution of characters with $d_\beta>0$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

I read the Alfaraj preprint on Manin's conjecture for equivariant compactifications of forms of G_a^n over function fields. The reader's report says ACCEPT, but the stress-test note is correct and it hits the main theorem.

The issue is in Proposition 8.3(ii). For d_α = m p^k, the proof claims the local character ψ_v(u x_α^{-d_α}) equals χ([x_α^{-d_α}, b)_v) for some b ∈ F_v^*. By Lemma 6.10, that equality forces (db/dπ_v)/b = u0. But the logarithmic derivative map b ↦ b'/b is not surjective onto units in characteristic p. I checked the coefficient recursion: for p=2 or p=3 and u0=1, the constant term forces b_1 = b_0, and then the next coefficient of b'/b is forced to be 1, not 0. No choice of higher coefficients fixes it. So the asserted existence of b fails already for a constant unit, and the reduction to the coprime case via Corollary 6.9 is not justified. The claimed bound for the local Fourier transform when p divides d_α is therefore unproven, and Corollary 8.4, Theorem 1.4, and the leading-constant theorem all rely on it.

This is not a minor gap; it is load-bearing. The paper's central claim is not established as written.

That said, the paper has real substance. Theorem 1.9 (Hasse principle and weak approximation for commutative unipotent groups admitting a smooth equivariant compactification) is a clean, independent result, and the Picard group computations in Section 3 look solid. The explicit P^{p-1} example in Section 10 is carefully worked out, including the verification of Assumption 1.2 and the explicit leading constant. The citation pattern is appropriate, and the reliance on Donlagić's preprint is a dependency, not a flaw.

The right reader is someone working on height zeta functions in positive characteristic; the Hasse-principle theorem is of independent interest. The paper deserves serious refereeing, but the appropriate outcome is major revision, not acceptance. The author should address Proposition 8.3(ii) directly, either with a corrected argument or by restricting the theorem to cases where the problematic characters do not arise.

Best,

[Your name]

Referee Report

1 major / 4 minor

Summary. The paper proves a version of Manin's conjecture for smooth equivariant compactifications of F-forms of G_a^n over global function fields. The main theorem gives absolute convergence and meromorphic continuation of the height zeta function under boundary assumptions (Assumption 1.2) and a character condition (Condition 9.1), identifies the poles of maximal order, and shows that the leading constant agrees with Peyre's prediction. It also proves that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation, and it computes in detail the example X=P^{p-1} with a natural F-wound group action. The argument follows the Chambert-Loir--Tschinkel height-zeta-function framework, adapted to the inseparable phenomena present in positive characteristic.

Significance. If the main theorem is correct, it is a substantive extension of the vector-group results of Chambert-Loir and Tschinkel to non-split forms of G_a^n, and it is the first treatment of the non-geometrically-reduced boundary components that necessarily appear when Pic(G) is non-trivial. The Hasse-principle and weak-approximation theorem (Theorem 1.9) and the explicit P^{p-1} example are valuable results in their own right. The proofs are detailed and the external benchmarks, such as comparison with Peyre's constant and with prior vector-group results, are the right ones. However, the local Fourier-transform estimate for nontrivial characters with p-divisible pole orders is not proved as written; since that estimate feeds directly into Corollary 8.4 and Theorem 1.4, the central claim is not yet established.

major comments (1)

[§8, Proposition 8.3(ii)] The reduction used for the case p dividing d_alpha is not justified. The displayed equality Ψ_{v,a}(x)=ψ_v(u x_alpha^{-d_alpha})=φ_{v,u_0}(x_alpha^{-d_alpha})=χ([x_alpha^{-d_alpha},b)_v) requires, by Lemma 6.10, an element b∈F_v^* with (db/dπ_v)/b=u_0. Such an element need not exist: for example, in F_2((π)) there is no b with b'/b=1. If v(b)=0, writing b=b_0+b_1π+... gives b'/b = b_1+b_1^2π+...; equality to 1 would force b_1=1 and then the π-coefficient is 1, not 0, while if v(b)≠0, b'/b has a pole. Moreover, even when such a b exists, Proposition 6.11 parametrizes additive characters y↦φ_{v,a(db/dπ)/b}(y) attached to linear forms a·x; it does not directly identify the multiplicative-type function x↦ψ_v(u x_alpha^{-d_alpha}). Corollary 6.9 concerns the identity [a^{p^m},b)=[a,b) for elements a of the field and does not supply the required reduction. Consequently the claimed vanishing estimate for p|d_alpha is unproven, and since Corollary 8.4 and Theorem 1.4 both rely on this estimate, the main theorem is not established as written.

minor comments (4)

[§8, Lemma 8.1] The proof of Lemma 8.1 contains a gap: after writing w=a+π^e z, the integral over z∈o_v is zero precisely when e−nd=−1, not for every e with nd/2≤e<nd. If e−nd≤−2, the character φ_v(u d a^{d-1}π^{e-nd}z) is trivial on o_v and the inner integral is 1. The statement is still correct and can be proved by choosing e=nd−1, but the proof as written should be corrected.
[Assumption 1.2(ii)] The notation in Assumption 1.2(ii)(2) and (ii)(4) is confusing: D_β(F_v)=∅ for the generic fibre, while the count in (4) concerns the special fibre of the model. The two objects should be distinguished notationally, for instance by writing D_β for the generic fibre and \mathcal{D}_β or \bar{D}_β for the reduction.
[§3, Proposition 3.8] In the proof of Proposition 3.8, the conclusion that ρ_α is an integer uses that the equality p_α ρ_α=ρ'_α, with ρ_α already an integer by the preceding divisor computation, forces p_α | ρ'_α. The proof should state this divisibility explicitly rather than passing directly from ρ_α=ρ'_α/p_α to the positivity conclusion.
[Abstract and §2] There are several typos: 'predicition' in the abstract, 'characterstic' in the title of Section 2, and 'compatﬁciation' and 'inseperable' elsewhere. These should be corrected during revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the height zeta function derivation is parameter-free and the leading constant is computed from independent geometric/adelic invariants; the skeptic's attack targets a proof gap, not a circular reduction.

full rationale

The paper's central derivation is self-contained and parameter-free. The height zeta function Z_λ(s) is defined directly from the height pairing (5.2), and its analytic properties are obtained via the Poisson summation formula (6.5) from Fourier transforms of the adelic height. No constant is fitted to the target asymptotic, and the asserted leading constant c_ρ is computed from Tamagawa measures, the effective cone constant α*(X), and |Pic(G)|, rather than defined as the residue being predicted. Assumption 1.2 and Condition 9.1 are structural hypotheses on the boundary divisor and on the characters contributing to the Poisson formula; they are not restatements of the conclusion, and the paper verifies them in the P^{p-1} example. The comparison with Peyre's prediction in Theorem 1.8 uses external results ([18, Theorem 4.5], [1, Corollary 4.14], [34, Theorem 1.1]) whose proofs do not presuppose the target asymptotic. There are no self-citations carrying a load-bearing argument: [12] is used only as the characteristic-zero model, and the acknowledgements identify the external source of Lemma 4.1. The skeptical attack on Proposition 8.3(ii) alleges an unjustified logarithmic-derivative surjectivity and an invalid reduction for local Fourier transforms when p divides d_α. If accurate, this is a correctness gap in the proof of the local Fourier bound, not a circularity: it does not exhibit an equation or fitted parameter that reduces the claimed prediction to its own input. Accordingly, no circular step is identified and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central theorem is conditional on Assumption 1.2 and Condition 9.1, which are hypotheses rather than derived facts. The proof also depends on several substantial external results from the literature (Donlagić, Rosengarten, Achet, Oesterlé, Chambert-Loir-Tschinkel). There are no fitted parameters or invented physical entities.

assumptions (8)

domain assumption Assumption 1.2: the boundary divisor D satisfies strict normal crossings on ∪_{α∉B}D_α; each D_β (β∈B) is geometrically irreducible, has no F_v-points, has section norms depending only on reduction, satisfies the point-count estimate #{x̄∈D_β(F_v): ‖s_β‖_v(x̄)=q_v^{-1}} = q_v^{dim X-1}+O(q_v^{dim…
Stated in Assumption 1.2 and used throughout §7.1-§7.4 to compute local Fourier transforms and prove Theorem 1.4. The paper verifies these properties only in the P^{p-1} example (§10).
ad hoc to paper Condition 9.1: every non-trivial character Ψ∈(G(A_F)/(G(F)+K))^∧ has a representative Ψ_a=ψ∘f_a whose divisor div(f_a)=E−∑d_αD_α has d_β=0 for all β∈B.
Introduced in §9 to apply Corollary 8.4 to all Fourier transforms; it is shown to hold only when Pic(G)=0, when G(A_F)=G(F)+K, or in the P^{p-1} example (Remark 9.2 and §10).
domain assumption [18, Theorem 4.5] (Donlagić, arXiv:2410.12127): for a homogeneous space Y of a commutative affine algebraic group over a global field, the Brauer-Manin obstruction B_ω(Y) is the only obstruction to weak approximation.
Used in the proof of Theorem 1.9 (§4.2, Proposition 4.4). The result is from a recent unrefereed preprint; the paper does not prove it.
domain assumption [34, Theorem 1.1] (Rosengarten): for a pseudo-reductive group G over a global function field, the Tamagawa number satisfies τ(G) = #Ext^1(G,G_m)/#X(G).
Used in §9.3 to compute the leading constant in Theorem 1.8.
domain assumption [1, Corollary 4.14] (Achet): Ext^1(G,G_m) ≅ Pic(G) for a commutative unipotent group G over a field of positive characteristic.
Used in §9.3 together with Rosengarten's formula to identify α^*(X).
domain assumption Oesterlé's structure theory for commutative unipotent groups in characteristic p: an F-form of G_a^n decomposes as a product G_a^m × W with W an F-wound group; F-wound groups have compact local point sets and split over a purely inseparable extension ([40, V.5, VI.2.1, VI.3.1]).
Used throughout §2 to describe forms, wound groups, and their compactifications, and in §10 for the example.
domain assumption Chambert-Loir-Tschinkel's analytic toolkit for vector-group compactifications: Poisson summation framework, height bounds ([12, Lemma 5.2]), local coordinate arguments ([12, Prop 10.2]), and their Theorem 0.1.
The paper explicitly models many arguments on [12] and cites it for several estimates; these are external results, though the paper redevelops much of the machinery.
standard math Standard analytic number theory and algebraic geometry background: Lang-Weil estimates, Chebotarev density for function fields, local class field theory symbols, Poisson summation on locally compact abelian groups, Rosenlicht's lemma, and the fact that global function fields have degree of…
Invoked in various sections (e.g., Lemma 4.3, Lemma 7.7, §6.2, Lemma 3.1); these are established results in the literature.

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Pith. "Pith review of Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$." pith.science (2026). https://pith.science/paper/ZKZFPNSX

@misc{pith2026250504562,
  author       = {Pith},
  title        = {Pith review of: Manin's Conjecture for Equivariant compactifications of forms of $\mathbbG_a^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKZFPNSX}},
  note         = {Machine review of arXiv:2505.04562}
}

abstract

We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over a global function field $F$, assuming some conditions on the boundary divisor. To verify that the leading constant agrees with Peyre's predicition we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. We study in detail the case of $\mathbb{P}^{p-1}$, where $p$ is the characteristic of $F$, viewed as a compactification of appropriate $F$-wound groups to illustrate new phenomena appearing in the function field setting.

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Forward citations

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For any finite abelian group G and global function field k, G-extensions with conductor q^M and prescribed local conditions satisfy an explicit asymptotic q^{aM} M^{b-1}, with local equidistribution governed by a Tama...

Reference graph

Works this paper leans on

42 extracted references · 41 canonical work pages · cited by 2 Pith papers

[1]

Achet, Picard group of the forms of the aﬃne line and of the additive g roup, Journal of Pure and Applied Algebra Volume 221(11) (2017), 2838-2860

R. Achet, Picard group of the forms of the aﬃne line and of the additive g roup, Journal of Pure and Applied Algebra Volume 221(11) (2017), 2838-2860

work page 2017
[2]

Achet, Unirational Algebraic Groups , 2019, available at https://hal.archives-ouvertes.fr/hal-02358528/document

R. Achet, Unirational Algebraic Groups , 2019, available at https://hal.archives-ouvertes.fr/hal-02358528/document

work page 2019
[3]

Bastos, Some results on the degree of imperfection of complete value d ﬁelds , Manuscripta Math 25 (1978), 315–322

G. Bastos, Some results on the degree of imperfection of complete value d ﬁelds , Manuscripta Math 25 (1978), 315–322

work page 1978
[4]

Batyrev and Y

V. Batyrev and Y. Manin, Sur le nombre des points rationnels de hauteur born´ ee des va ri´ et´ es alg´ ebriques, Math. Ann. 286 (1990), 27-43

work page 1990
[5]

Becker and S

M. Becker and S. MacLane, The minimum number of generators for in- separable extensio ns, Bull. Amer. Math. Soc.46 (1940), 182-186

work page 1940
[6]

Borel, Linear algebraic groups , Second enlarged edition, Gra

A. Borel, Linear algebraic groups , Second enlarged edition, Gra. Texts Math. 126, Springer, 1991

work page 1991
[7]

Bosch, W

S. Bosch, W. Lutkebohmert, M. Raynaud, N´ eron Models, Ergebnisse der Math. Springer Heidelberg, 21, 1990

work page 1990
[8]

Boucksom, W

S. Boucksom, W. Gubler and F. Martin, Non-Archimedean volumes of metrized nef line bundles , ´Epijournal de G´ eom´ etrie Alg´ ebrique, 2020

work page 2020

Show all 42 references

[9]

Browning, An overview of Manin’s conjecture for del Pezzo surfaces , In Analytic number theory, volume 7 of Clay Math

T. Browning, An overview of Manin’s conjecture for del Pezzo surfaces , In Analytic number theory, volume 7 of Clay Math. Proc., pages 39–55. Amer. Math. Soc., Prov idence, RI, 2007

2007
[10]

Brion, On Linearization of Line Bundles , J

M. Brion, On Linearization of Line Bundles , J. Math. Sci. Univ. Tokyo 22 (2015), 113–147

2015
[11]

Chambert-Loir, Y

A. Chambert-Loir, Y. Tschinkel, Igusa integrals and volume asymptotics in analytic and adel ic geometry, Conﬂuences Math. 2(3) (2010), 351–429

2010
[12]

Chambert-Loir, Y

A. Chambert-Loir, Y. Tschinkel, On the distribution of points of bounded height on equivaria nt compactiﬁcations of vector groups , Invent. math. 148 (2002), 421–452

2002
[13]

Chambert-Loir, Lectures on height zeta functions: At the conﬂuence of algeb raic geometry, algebraic number theory, and analysis , MSJ Memoirs 21 (2010), 17-49

A. Chambert-Loir, Lectures on height zeta functions: At the conﬂuence of algeb raic geometry, algebraic number theory, and analysis , MSJ Memoirs 21 (2010), 17-49

2010
[14]

Colliot-Th´ el` ene, A

J. Colliot-Th´ el` ene, A. Skorobogatov, The Brauer–Grothendieck Group , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Su rveys in Mathematics, Springer, 2021

2021
[15]

Conrad, O

B. Conrad, O. Gabber, G. Prasad, Pseudo-reductive Groups , Cambridge Univ. Press (2nd edition), 2015

2015
[16]

Conrad, Finiteness theorems for algebraic groups over function ﬁel ds, Compositio Math

B. Conrad, Finiteness theorems for algebraic groups over function ﬁel ds, Compositio Math. 148 (2012), 555-639

2012
[17]

J. Denef. On the degree of Igusa’s local zeta function , Amer. J. Math. 109 (1987), 991–1008

1987
[18]

MANIN’S CONJECTURE FOR COMPACTIFICATIONS OF FORMS OF Gn a 53

Azur ¯Donlagi´ cBrauer-Manin obstructions for homogeneous spaces of commu tative aﬃne algebraic groups over global ﬁelds , arXiv:2410.12127 (2024). MANIN’S CONJECTURE FOR COMPACTIFICATIONS OF FORMS OF Gn a 53

2024
[19]

Franke, Y

J. Franke, Y. Manin, Y. Tschinkel, Rational points of bounded height on Fano varieties . Invent. Math. 95 (1989), 421–435

1989
[20]

Herrero, T

S. Herrero, T. Mart ´ ınez, P. Montero, Counting rational points on Hirzebruch-Kleinschmidt varieties over global function ﬁelds , arXiv:2408.07631 (2024)

2024 arXiv
[21]

Illusie and M

L. Illusie and M. Temkin, Expos´ e X. Gabber’s modiﬁcation theore m (log smooth case), Ast´ erisque 363-364 (2014), 167–212. Travaux de Gabber sur l’u niformisation locale et la cohomologie ´ etale des sch´ emas quasi-excellents

2014
[22]

Kambayashi, M

T. Kambayashi, M. Miyanishi, M. Takeuchi, Unipotent Algebraic Groups , Lecture Notes in Mathematics, vol 414. Springer, Berlin, Heidelberg (1974)

1974
[23]

Lang and A

S. Lang and A. Weil, Number of points of varieties in ﬁnite ﬁelds , Amer. J. Math. 76 (1954), 819–827

1954
[24]

MacLane, The minimum number of generators for in- separable exte nsions, Bull

Qing Liu, Algebraic Geometry and Arithmetic Curves , Oxford University Press, 2006. MacLane, The minimum number of generators for in- separable exte nsions, Bull. Amer. Math. Soc. 46 (1940), 182-186

1940
[25]

Manin and A

Y. Manin and A. Panchishkin, Number theory I. Introduction to number theory , Springer, Berlin, 1995

1995
[26]

Mumford, J

D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory . Third edition, Ergeb. Math. Grenzgeb. 34(2), Springer-Verlag, Berlin, 1994

1994
[27]

Peyre, Hauteurs et mesures de Tamagawa sur les vari´ et´ es de Fano , Duke Math

E. Peyre, Hauteurs et mesures de Tamagawa sur les vari´ et´ es de Fano , Duke Math. J. 79(1) (1995), 101–218

1995
[28]

Peyre, Points de hauteur born´ ee sur les vari´ et´ es de drapeaux en caract´ eristique ﬁnie, Acta Arithmetica 152(2) (1995), 185–216

E. Peyre, Points de hauteur born´ ee sur les vari´ et´ es de drapeaux en caract´ eristique ﬁnie, Acta Arithmetica 152(2) (1995), 185–216

1995
[29]

Pieropan, A

M. Pieropan, A. Smeets, S. Tanimoto, A. V´ arilly-Alvarado, Campana points of bounded height on vector group compactiﬁcations , Proc. London Math. Soc. 123(3) (2021), 57–101

2021
[30]

Poonen, Rational points on varieties , Graduate Studies in Mathematics 186, American Mathematical Society, Providence, 2017

B. Poonen, Rational points on varieties , Graduate Studies in Mathematics 186, American Mathematical Society, Providence, 2017

2017
[31]

Rosen, Number Theory in Function Fields , Springer-Verlag, New York, 2002

M. Rosen, Number Theory in Function Fields , Springer-Verlag, New York, 2002

2002
[32]

Rosenlicht, Toroidal algebraic groups , Proceedings of the American Mathematical Society 12 (6) (1961), 984-988

M. Rosenlicht, Toroidal algebraic groups , Proceedings of the American Mathematical Society 12 (6) (1961), 984-988

1961
[33]

Rosengarten, Pathological Behavior of Arithmetic Invariants of Unipote nt Groups , Algebra and Number Theory 15(7) (2021), 1593-1626

Z. Rosengarten, Pathological Behavior of Arithmetic Invariants of Unipote nt Groups , Algebra and Number Theory 15(7) (2021), 1593-1626

2021
[34]

Rosengarten, Tamagawa Numbers And Other Invariants of Pseudo-reductive Groups Over Global Function Fields , Algebra and Number Theory 15(8) (2021), 1865–1920

Z. Rosengarten, Tamagawa Numbers And Other Invariants of Pseudo-reductive Groups Over Global Function Fields , Algebra and Number Theory 15(8) (2021), 1865–1920

2021
[35]

Rosengarten, N

Z. Rosengarten, N. Tˆ an, N. Thang,On The Galois And Flat Cohomology Of Unipotent Algebraic Groups Over Local And Global Function Fields II , Michigan Mathematical Journal Advance Publication (2023)

2023
[36]

Serre, Local Fields, Graduate Texts in Mathematics, vol 67, 1979

J.-P. Serre, Local Fields, Graduate Texts in Mathematics, vol 67, 1979

1979
[37]

Salberger, Tamagawa measures on universal torsors and points of bounde d height on Fano varieties, Ast´ erisque251 (1998), 91–258

P. Salberger, Tamagawa measures on universal torsors and points of bounde d height on Fano varieties, Ast´ erisque251 (1998), 91–258. Nombre et r´ epartition de points de hauteur bor n´ ee (Paris, 1996)

1998
[38]

Tits, Lectures on algebraic groups , notes by P

J. Tits, Lectures on algebraic groups , notes by P. Andr´ e and D. Winter. Yale University, 1968

1968
[39]

The Stacks Project Authors, Stacks Project, 2024

2024
[40]

Oesterl´ e, Nombres de Tamagawa et groupes unipotents en caract´ eristi que p , Inventiones mathematicae 78(1) (1984), 13–88

J. Oesterl´ e, Nombres de Tamagawa et groupes unipotents en caract´ eristi que p , Inventiones mathematicae 78(1) (1984), 13–88

1984
[41]

Weil, Basic Number Theory , Springer Berlin Heidelberg, 1973

A. Weil, Basic Number Theory , Springer Berlin Heidelberg, 1973

1973
[42]

Weil, Adeles and algebraic groups , Progr

A. Weil, Adeles and algebraic groups , Progr. Math., no. 23, Birkh¨ auser, 1982. Abdulmuhsin Alfaraj, Department of Mathematical Sciences , University of Bath, Claverton Down, Bath, BA2 7AY, UK

1982

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Achet, Picard group of the forms of the aﬃne line and of the additive g roup, Journal of Pure and Applied Algebra Volume 221(11) (2017), 2838-2860

R. Achet, Picard group of the forms of the aﬃne line and of the additive g roup, Journal of Pure and Applied Algebra Volume 221(11) (2017), 2838-2860

work page 2017

[2] [2]

Achet, Unirational Algebraic Groups , 2019, available at https://hal.archives-ouvertes.fr/hal-02358528/document

R. Achet, Unirational Algebraic Groups , 2019, available at https://hal.archives-ouvertes.fr/hal-02358528/document

work page 2019

[3] [3]

Bastos, Some results on the degree of imperfection of complete value d ﬁelds , Manuscripta Math 25 (1978), 315–322

G. Bastos, Some results on the degree of imperfection of complete value d ﬁelds , Manuscripta Math 25 (1978), 315–322

work page 1978

[4] [4]

Batyrev and Y

V. Batyrev and Y. Manin, Sur le nombre des points rationnels de hauteur born´ ee des va ri´ et´ es alg´ ebriques, Math. Ann. 286 (1990), 27-43

work page 1990

[5] [5]

Becker and S

M. Becker and S. MacLane, The minimum number of generators for in- separable extensio ns, Bull. Amer. Math. Soc.46 (1940), 182-186

work page 1940

[6] [6]

Borel, Linear algebraic groups , Second enlarged edition, Gra

A. Borel, Linear algebraic groups , Second enlarged edition, Gra. Texts Math. 126, Springer, 1991

work page 1991

[7] [7]

Bosch, W

S. Bosch, W. Lutkebohmert, M. Raynaud, N´ eron Models, Ergebnisse der Math. Springer Heidelberg, 21, 1990

work page 1990

[8] [8]

Boucksom, W

S. Boucksom, W. Gubler and F. Martin, Non-Archimedean volumes of metrized nef line bundles , ´Epijournal de G´ eom´ etrie Alg´ ebrique, 2020

work page 2020