Finite abelian subgroups of algebraic groups

Danny Ofek; Federico Scavia; Zinovy Reichstein

REVIEW 1 major objections 4 minor 17 references

Finite abelian subgroups of algebraic groups

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For any affine algebraic group G over an algebraically closed field, every finite abelian subgroup A whose order is not divisible by the characteristic meets some maximal torus T in a subgroup of index dividing the Grothendieck torsion…

desk verdict Strong paper that answers the Totaro–Wang question and computes the E8 variant t2=60; the only real soft spot is a terse citation in Proposition 5.2 that should be checked by a referee. read the letter →

arxiv 2608.01595 v1 pith:4EJOGLPL submitted 2026-08-03 math.AG math.GR

Danny Ofek , Zinovy Reichstein , Federico Scavia This is my paper · ORCID

classification math.AGmath.GR MSC 20G0720G1020G4112J10

keywords finiteabeliansubgroupsmaximaltorusGrothendiecktorsionindexprimesE8torsorsiteratedLaurentseriesfieldsgenusonecurves

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 proves a uniform structural bound on finite abelian subgroups of algebraic groups. Over an algebraically closed field, for any affine group G and any finite abelian subgroup A whose order is not divisible by the field characteristic, some maximal torus T of G satisfies $[A : A\cap T] \mid t(G)$, where $t(G)$ is the Grothendieck torsion index. Since $t(G)$ is a fixed integer computed from the degrees of fields needed to split $G$-torsors, the theorem reduces a question about arbitrary finite abelian subgroups to a computation of one invariant. This is the engine behind the paper's applications: a positive answer to a question about $G$-torsors over iterated Laurent series fields, the value $t_2(E_8)=60$ for an $E_8$-group in characteristic zero, and obstructions to splitting certain torsors by genus one curves.

What carries the argument

The main machinery is the depth of $A$, the greatest common divisor of the indices $[A : A\cap T]$ as $T$ ranges over maximal tori, together with the Grothendieck torsion index $t(G)$, which is the least common multiple of indices of all $G$-torsors over all field extensions and can be computed as the index of a single versal torsor. The pivotal structural proposition shows that if $A$ is a product of pairwise-coprime-order subgroups and each factor is toral, then $A$ is toral; its proof needs, for a finite group acting on the connected centralizer of one factor, a maximal torus invariant under the action. Around this core, the paper connects torsors over iterated Laurent series fields with Galois groups of their splitting fields, and uses a classification of maximal finite abelian subgroups of $E_8$ together with a correspondence between such subgroups and self-dual binary codes to compute depths.

What would settle it

To refute the main theorem, exhibit an algebraically closed field $k$, an affine $k$-group $G$, and a finite abelian subgroup $A$ with $\operatorname{char}(k)\nmid |A|$ such that every maximal torus $T$ of $G$ satisfies $[A : A\cap T]\nmid t(G)$. A targeted check is to test the semisimple reduction directly: find a finite group acting on a reductive group whose invariant maximal torus in the quotient by the center does not lift to an invariant maximal torus of the original group.

Watch

Extended reading notes

Core claim

The central claim is that non-toral parts of finite abelian subgroups are controlled by torsion. Concretely, if $A\subset G$ is finite abelian and $\operatorname{char}(k)\nmid |A|$, then $A$ has a toral subgroup $A\cap T$ whose index divides the Grothendieck torsion index $t(G)$; moreover, one can choose $T$ so that this index is exactly the depth of $A$, the greatest common divisor of all such indices. The paper obtains this by extending an earlier characteristic-zero result, which treated only $p$-subgroups, to all prime-to-characteristic finite abelian subgroups, and by working without resolution of singularities so that positive characteristic is covered. Under the mild 'good characteristic' assumption, the paper also identifies the torsion-index variant $t_2(G)$, defined through torsors over iterated Laurent series fields, with the depth invariant $t_3(G)$.

Load-bearing premise

The load-bearing step is the reduction, in the proof of the key proposition, from a reductive connected centralizer to its semisimple quotient when seeking a maximal torus invariant under the action of a complementary finite subgroup; if that reduction fails to produce an invariant maximal torus, the proof of Theorem 1.1 collapses.

Editorial extensions

If this is right

For a prime $p$ that does not divide $t(G)$, every finite abelian $p$-subgroup of $G$ is toral; this makes the classical characterization of torsion primes by non-toral elementary abelian subgroups quantitative.
Over $k((t_1))\cdots((t_r))$, every $G$-torsor with a zero-cycle of degree $d$ has a closed point of degree dividing $d$ whenever $G$ is smooth affine and the characteristic is good for $G$.
For a group of type $E_8$ in characteristic zero, the iterated-Laurent-series torsion index is $t_2(E_8)=60$, so the 'optimistic hypothesis' on splitting fields of $E_8$-torsors holds for this variant even though the full Grothendieck torsion index is $26325$.
A torsor induced from an elementary abelian $p$-subgroup of sufficiently large non-toral rank cannot be split by any torsor under a $d$-dimensional abelian variety, and in particular not by a genus one curve; this yields explicit $E_8$-torsors that cannot be split by a genus one curve.

Reading between the lines

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

The bound $[A : A\cap T]\mid t(G)$ is probably sharp only for $p$-primary subgroups at torsion primes; for composite-order $A$ the realized indices may be much smaller, and a prime-by-prime refinement of the depth invariant could sharpen the applications to torsor splitting.
The equality $t_2(G)=t_3(G)$ in good characteristic suggests that the full Grothendieck torsion index may sometimes be recovered by looking only at torsors over iterated Laurent series fields; testing whether $t_2(G)=t(G)$ for reductive groups with no small torsion primes would clarify how much of the classical invariant this new viewpoint captures.
The use of self-dual binary codes in the $E_8$ depth computation indicates that the same path could compute $t_2(\operatorname{Spin}_n)$ and relate splitting obstructions for $\operatorname{Spin}_n$-torsors to coding-theoretic invariants such as minimum weight.
Because the main theorem excludes finite abelian subgroups whose order is divisible by the field characteristic, any analogue in positive characteristic would require a separate invariant in place of $t(G)$; the paper leaves that boundary unexplored.

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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

Worth your time. The paper answers the Totaro–Wang question (Theorem 1.1): for an affine k-group over an algebraically closed field, every finite abelian prime-to-p subgroup is close to toral, with the defect dividing the Grothendieck torsion index. That alone justifies publication. It also computes t2(E8)=60, saving a version of Tits' “hypothèse optimiste,” and extends the genus-one splitting obstruction from [RS26] to a general setting.

The proofs are serious rather than merely sketched. The E8 section checks all eight Levi subgroups and does the depth computations with explicit traces; the spin/code case is concrete. The chain from Proposition 3.1 through Lemma 4.3 and Corollary 5.3 is mostly self-contained and does not reduce to the authors' previous work. The reuse of [RS26] and [RY01b] is legitimate; the new theorems are genuinely new.

The one point I would flag is in Proposition 5.2. The proof invokes [SS70, Theorem 5.16] to get a maximal torus of C_G(A1)^0 stable under the action of the finite group A2. The theorem as usually cited is stated for semisimple groups, and the footnote's reduction to the semisimple case by quotienting the center is plausible: the preimage of a maximal torus is a torus. But the paper doesn't verify that [SS70, Thm 5.16] covers a finite group of automorphisms, as opposed to a single automorphism. If it covers only a single automorphism, one needs a short extra argument to get the whole group A2 stabilized. This is a local presentation gap, not evidence that the theorem is false; the authors also cite [CGP25, Prop 5.10] as an alternative. I'd want a referee to check that passage before publication.

No other soft spot matches it. The circularity burden is low: Theorem 1.1 is proved via the new depth/cohomology machinery, not by assuming what it proves.

This is for specialists in algebraic groups, torsors, and essential dimension. It deserves a serious referee and will be read closely. I recommend sending it to peer review with a request to spell out the [SS70] step.

Best,

Referee Report

1 major / 4 minor

Summary. The paper studies finite abelian k-subgroups A of affine algebraic groups over an algebraically closed field k, with char(k) not dividing |A|. Its main theorem (Theorem 1.1) asserts that for every such A there exists a maximal torus T of G such that the index [A : A∩T] divides the Grothendieck torsion index t(G). The proof introduces the depth invariant t3(G), shows t2(G)=t3(G) under mild characteristic assumptions, and proves a structural result (Proposition 5.2) saying that products of toral subgroups of pairwise coprime orders are toral. The applications are: a positive answer to Totaro's question for torsors over iterated Laurent series fields (Theorem 1.5), a variant of Tits' optimistic hypothesis for E8 with t2(E8)=60 (Theorem 1.3), and obstructions to splitting torsors by genus-one curves (Theorem 1.8). Section 7 contains detailed computations of depths of maximal finite abelian subgroups of E8, including a code-theoretic construction via Wood's C16 code and an explicit trace computation for the μ4^3×μ2^2 case.

Significance. If Theorem 1.1 is correct, it answers a question of Totaro and Wang and provides a uniform bound, depending only on G, on how far a finite abelian prime-to-p subgroup is from being toral. The applications are substantial, and the paper is unusually careful: the eight maximal Levi subgroups of E8 are checked, the trace computations in Claim 7.3 are exhibited, and the construction of the toral subgroup in Proposition 6.2(e) via the 8-by-16 generator matrix for the self-dual code C16 is explicit and verifiable. The main risk is a cited-theorem gap in Proposition 5.2; provided that gap is repaired, the paper is a strong contribution.

major comments (1)

[§5, Proposition 5.2 and footnote 1] The proof of Proposition 5.2 invokes [SS70, Theorem 5.16] to obtain a maximal torus of H = C_G(A1)^0 that is invariant under the action of the finite abelian group A2. The cited theorem is stated for semisimple groups, while H is merely reductive. The footnote asserts that one can reduce to the semisimple case by modding out by the center, and cites [CGP25, Proposition 5.10] as an alternative, but the paper gives no proof that an A2-stable maximal torus of H/Z(H) lifts to an A2-stable maximal torus of H, and it does not state whether [SS70, Theorem 5.16] covers a finite abelian group of automorphisms or only a single semisimple automorphism. If iteration over elements of A2 is needed, each step requires a statement about invariant maximal tori in fixed-point subgroups of semisimple automorphisms, which is not automatic. Since Proposition 5.2 feeds Corollary 5.3 and hence Theorem 1.1, this is load-bearing; please supply a complete argument or a precise statement and reference covering the reductive case with a finite abelian automorphism group.

minor comments (4)

[§7.2 and footnote 3] When citing [RY01b, Proposition 5.3] for a self-centralizing subgroup isomorphic to μ2^8 of depth 4, the different convention for depth in [RY01b] should be spelled out immediately in the text, since the footnote already notes that the present paper uses the index rather than the exponent.
[§1.5 and §7.5] Please ensure that the notation for the non-elementary maximal finite abelian subgroups of E8 is unambiguous: the group written as μ_3^6 in the introduction and in Proposition 6.2(c) appears to be μ_6^3 (three copies of cyclic group of order 6), and the proof in §7.5 only makes sense with that reading.
[§3, proof of Theorem 1.2] The assertion that the finite extension F/k_r is isomorphic to k_r over k, cited as [GR09, Corollary 5.4], is surprising and is used to apply Lemma 2.8 over F; a brief explanation or a direct quotation of the cited statement would improve readability.
[§7.4, Lemma 7.1(2)] The displayed conclusion of Lemma 7.1(2) is visually garbled in the typeset text; it should state that the fixed-point subalgebra of the adjoint action of A_C on so_{2n} is zero.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 and its applications are derived from new cohomological arguments and independent published theorems; the only flagged issue is an unproved reductive-to-semisimple reduction in a footnote, which is a correctness risk, not circularity.

full rationale

The paper's central derivation is self-contained rather than circular. Theorem 1.1 is deduced from Corollary 5.3 and Lemma 4.3(2), neither of which is a fitted parameter renamed as a prediction. Corollary 5.3 follows from Proposition 5.2, whose proof uses a genuine cohomological argument (twisting, Serre's question for tori, and [GS18, Lemma 3.1]) together with the independent theorem [SS70, Theorem 5.16]. Lemma 4.3(2) proves the equality of the index of an induced torsor with the depth of the finite abelian subgroup, and its converse invokes Theorem 1.2(1), which was already proved in Section 3 via Proposition 3.1; this is not circular because Theorem 1.2(1) does not depend on Lemma 4.3. The equality t2(G) = t3(G) in Proposition 4.2 is likewise a proved equivalence, not a definitional shortcut. The E8 application is supported by the independent classification of [DE17], code-theoretic arguments, and explicit trace computations in Section 7; the depth values are not calibrated to force t2(E8) = 60. The paper cites prior work by its own authors, notably [RS26], [RY01a], [RY01b], and [RY00], but these are specific published theorems used as ingredients, and the present paper's main claims do not reduce to them by construction. The one manuscript passage that warrants flagging is footnote 1 to Proposition 5.2: it asserts without proof a reduction from the reductive group C_G(A1)^0 to the semisimple case when applying [SS70, Theorem 5.16], citing [CGP25, Proposition 5.10] as an alternative. This is a potential correctness gap in the proof of Proposition 5.2, and hence of Theorem 1.1, but it is not a circularity: it is a missing verification of a standard-looking lifting statement, not an input fed back as the output. Accordingly, no circular steps are recorded and the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or algebraic entities; it defines new numerical invariants t_2(G), t_3(G), and depth(A), which are functions of the group and subgroup, not additional structure. The free-parameter list is empty because there is no data fitting. The axioms listed are the externally cited results and explicit hypotheses that carry the load.

assumptions (5)

standard math Standard theorems of algebraic groups, Galois cohomology, and the structure theory of reductive groups (Grothendieck's torsion index, Steinberg's classification of torsion primes, Hilbert Theorem 90, SGA3 results on Levi subgroups and twisting).
Used throughout; explicitly cited in Sections 2, 3, 5, and 6.
domain assumption Draper-Elduque classification [DE17] of maximal finite abelian subgroups of E8 up to conjugacy: seven types, uniquely determined by isomorphism type.
Section 6 and 7 (Proposition 6.2 and its proof); this external classification is a black box for the depth computations.
domain assumption [SS70, Theorem 5.16]: existence of a maximal torus of a semisimple group normalized by a finite group of automorphisms, extended to the reductive centralizer C_G(A_1)^0 by modding out by the center.
Proof of Proposition 5.2, page 14, footnote 1; the extension to reductive groups is asserted with a short justification and an alternative reference [CGP25, Prop 5.10].
domain assumption [GR09, Corollaries 5.4 and 6.4]: for extensions F or E of the iterated Laurent series field k_r, the fields are isomorphic over k, allowing transfer of Galois group structure.
Conclusion of the proof of Theorem 1.2 in Section 3; used to identify Gal(F/E) as a semidirect product and to transfer splitting.
domain assumption Good characteristic hypothesis (Definition 2.9) condition char(k) does not divide |W(G')|·[G:G^0], used in Lemma 2.10 to reduce arbitrary torsors to finite abelian subgroups, and in Proposition 4.2 to equate t_2 and t_3.
Explicit hypothesis of Lemma 2.10, Proposition 4.2, and Theorem 1.5; not needed for the central Theorem 1.1.

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Pith. "Pith review of Finite abelian subgroups of algebraic groups." pith.science (2026). https://pith.science/paper/4EJOGLPL

@misc{pith2026260801595,
  author       = {Pith},
  title        = {Pith review of: Finite abelian subgroups of algebraic groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EJOGLPL}},
  note         = {Machine review of arXiv:2608.01595}
}

abstract

Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A/(A \cap T)$ is ``small'' in a suitable sense. As applications of these results, we (i) give a positive answer to a question of Totaro for $G$-torsors over fields $k_r = k((t_1))((t_2)) \ldots ((t_r))$ of iterated Laurent series, (ii) prove a variant of the ``hypoth\`ese optimiste'' of Tits about splitting fields of $E_8$-torsors, and (iii) show that certain torsors over $k_r$ cannot be split by the function field of a genus $1$ curve.

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Works this paper leans on

17 extracted references · 17 canonical work pages

[2]

Proposition 6.2

In particular, up to conjugacy, a maximal ﬁnite abelian subgroup /u1D434of/u1D4388 is uniquely determined by its isomorphism type. Proposition 6.2. Assume char(/u1D458) = 0. Let /u1D434be a maximal ﬁnite abelian subgroup of a split group of type /u1D4388. (a) If/u1D434≃/u1D7079 2, then depth(/u1D434) = 2, (b) If/u1D434≃/u1D7078 2, then depth(/u1D434) = 4,...

work page
[3]

Under this assumption, both parts reduce to the following

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc1.2 We ﬁrst prove /T_heorem1.2 in the special case, where /u1D43F=/u1D458/u1D45F. Under this assumption, both parts reduce to the following. Proposition 3.1. Let/u1D43Abe an algebraic /u1D458-group and let /u1D434⊂/u1D43Abe a ﬁnite abelian /u1D458-subgroup of rank/u1D45F. Assume that char(/u1...

work page
[4]

As we pointed out in the Introduction, we have (4.1) /u1D4613(/u1D43A)| /u1D4612(/u1D43A)| /u1D461(/u1D43A)

V /a.sc/r.sc/i.sc/a.sc/t.sc/i.sc/o.sc/n.sc/s.sc /o.sc/n.sc /t.sc/h.sc/e.sc /t.sc/o.sc/r.sc/s.sc/i.sc/o.sc/n.sc /i.sc/n.sc/d.sc/e.sc/x.sc In (1.1)-(1.3), we deﬁned the torsion indices /u1D461(/u1D43A),/u1D4612(/u1D43A) and/u1D4613(/u1D43A) for any aﬃne/u1D458-group/u1D43A. As we pointed out in the Introduction, we have (4.1) /u1D4613(/u1D43A)| /u1D4612(/u1...

work page
[5]

/T_his notion was introduced by Serre in [ Ser58] (reprinted in [ Ser01]), where he also showed that special /u1D458-groups are smooth, aﬃne and connected

P/r.sc/o.sc/o.sc/f.sc/s.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc/s.sc1.1 /a.sc/n.sc/d.sc1.5 Recall that an algebraic /u1D458-group/u1D43Ais called special if/u1D43B1(/u1D43E,/u1D43A) = 1 for every ﬁeld extension /u1D43E//u1D458. /T_his notion was introduced by Serre in [ Ser58] (reprinted in [ Ser01]), where he also showed that special /u1D458-groups...

work page
[6]

Lemma 6.1

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc1.3 Our proof is based on the following lemma. Lemma 6.1. Let/u1D43Abe a semisimple/u1D458-group. For any proper parabolic subgroup/u1D443⊂/u1D43A, let/u1D43F/u1D443⊂/u1D443 denote a Levi subgroup. /T_he index/u1D4613(/u1D43A) is the least common multiple of the depths of all maximal ﬁnite ab...

work page
[7]

(See [ Tot05a] for torsion indices of /u1D438sc 6 and/u1D438sc 7 .) We conclude that ( 6.4) holds in each of the cases (i) - (viii), as we wanted to show

= 12, respectively. (See [ Tot05a] for torsion indices of /u1D438sc 6 and/u1D438sc 7 .) We conclude that ( 6.4) holds in each of the cases (i) - (viii), as we wanted to show. □ Remark 6.3. Let/u1D43Abe an aﬃne group over an algebraically closed ﬁeld /u1D458. Recall that /u1D461(/u1D43A) is deﬁned as the least common multiple of ind(/u1D44B), as/u1D44Brang...

work page
[8]

Let/u1D458be a ﬁeld of characteristic zero, and let /u1D43Abe a semisimple/u1D458-group of type /u1D4388

D/e.sc/p.sc/t.sc/h.sc/s.sc /o.sc/f.sc /t.sc/h.sc/e.sc /m.sc/a.sc/x.sc/i.sc/m.sc/a.sc/l.sc /f.sc/i.sc/n.sc/i.sc/t.sc/e.sc /a.sc/b.sc/e.sc/l.sc/i.sc/a.sc/n.sc /s.sc/u.sc/b.sc/g.sc/r.sc/o.sc/u.sc/p.sc/s.sc /o.sc/f.sc/u1D4388 In this section we prove Proposition 6.2. Let/u1D458be a ﬁeld of characteristic zero, and let /u1D43Abe a semisimple/u1D458-group of ty...

work page
[9]

We denote the /u1D456-th coordinate of a word /u1D464∈/u1D436by/u1D464/u1D456

Elements of /u1D436are called words. We denote the /u1D456-th coordinate of a word /u1D464∈/u1D436by/u1D464/u1D456. /T_heHamming weight of/u1D464∈ F/u1D45B 2 is the number of non-zero coordinates wt(/u1D464) = /barex /barex{1≤/u1D456≤/u1D45B:/u1D464/u1D456= 1} /barex /barex. We use the notation (F/u1D45B 2)0 ⊂ F/u1D45B 2 for the subspace consisting of all...

work page

Show all 17 references

[10]

/T_his shows that (2) holds, ﬁnishing the proof

/T_herefore/u1D434= ˜/u1D434is maximal. /T_his shows that (2) holds, ﬁnishing the proof. □ 7.5. Case (f). /T_he following lemma will be used in the proofs of parts (f) and (g) of P roposi- tion 6.2. Lemma 7.2. Let/u1D43Bbe a semisimple algebraic group over a ﬁeld /u1D458of cha...
[11]

= det(1+/u1D465/u1D435/u1D45F,/u1D460)2+ 3 det(1+/u1D4652/u1D4352 /u1D45F,/u1D460). Set/u1D439/u1D45F,/u1D460(/u1D465) := det(1+/u1D465/u1D435/u1D45F,/u1D460)2+ 3 det(1+/u1D4652/u1D4352 /u1D45F,/u1D460), so that the previous equation implies ∑ /u1D452,/u1D453 [/u1D465/u1D45A] ...
[12]

=[/u1D465/u1D45A]/u1D439/u1D45F,/u1D460(/u1D465). /T_he characteristic polynomials of the monomial matrices/u1D435/u1D45F,/u1D460=/u1D701/u1D45F+/u1D460/u1D443/u1D45F/u1D446/u1D460give the following table: (/u1D45F,/u1D460) det(1+/u1D465/u1D435/u1D45F,/u1D460) det(1+/u1D4652/u...
[13]

=              2, (/u1D45F,/u1D460) =(0, 0), −2, (/u1D45F,/u1D460) =(2, 0) or(0, 2), 2, (/u1D45F,/u1D460) =(2, 2), 0, otherwise. /T_herefore, ( 7.10) yields∑ /u1D45F,/u1D460,/u1D452,/u1D453 tr(/u1D454/u1D45F,/u1D460,/u1D452,/u1D453|/u1D448⊗∧ 2/u1D449) = 2· 40+(− 2...
[14]

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc1.8 Deﬁnition 8.1. For a prime/u1D45Dand a ﬁnite group Γ, the/u1D45D-rank of Γ, denoted by rank/u1D45D(Γ), is the largest integer/u1D45F/greater⋊requalslant 0 such that Γ has a subgroup isomorphic to(Z//u1D45DZ)/u...

1991 arXiv
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Ramiﬁcation in Algebra and Geometry at Emory

21 28 DANNY OFEK, ZINOVY REICHSTEIN, AND FEDERICO SCA VIA [RS26] Zinovy Reichstein and Federico Scavia. Brauer classes not split by genus one curves. Invent. Math. , published on line, July 2026. h/t_tps://doi.org/10.1007/s00222-026-01437-1. 3, 4, 5, 25 [RT20] Zinovy Reichstei...

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3, 8 [Tit92b] Jacques Tits

92e ann ´ee. 3, 8 [Tit92b] Jacques Tits. Sur les degr ´es des extensions de corps d ´eployant les groupes alg ´ebriques simples. C. R. Acad. Sci. Paris S ´er. I Math., 315(11):1131–1138, 1992. 3 [Tot04] Burt Totaro. Spli/t_ting ﬁelds for /u1D4388-torsors. Duke Math. J., 121(3)...

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Espaces ﬁbr ´es alg´ebriques, expos´e 1

2, 3, 19, 26 [Ser58] Jean-Pierre Serre. Espaces ﬁbr ´es alg´ebriques, expos´e 1. In S´eminaire C. Chevalley; 2e ann´ee. Anneaux de Chow et applications, pages iii+134 pp. (mimeographed). Secr ´etariat math´ematique, 11 rue Pierre Curie, Paris, 1958. 13 [Ser95] Jean-Pierre Serr...

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2 [Woo89] Jay A. Wood. Spinor groups and algebraic coding theory. J. Combin. /T_heory Ser. A, 51(2):277–313, 1989. 3, 19, 20 FINITE ABELIAN SUBGROUPS OF ALGEBRAIC GROUPS 29 D/e.sc/p.sc/a.sc/r.sc/t.sc/m.sc/e.sc/n.sc/t.sc /o.sc/f.sc M/a.sc/t.sc/h.sc/e.sc/m.sc/a.sc/t.sc/i.sc/c.sc...

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Pith tools

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

[1] [2]

Proposition 6.2

In particular, up to conjugacy, a maximal ﬁnite abelian subgroup /u1D434of/u1D4388 is uniquely determined by its isomorphism type. Proposition 6.2. Assume char(/u1D458) = 0. Let /u1D434be a maximal ﬁnite abelian subgroup of a split group of type /u1D4388. (a) If/u1D434≃/u1D7079 2, then depth(/u1D434) = 2, (b) If/u1D434≃/u1D7078 2, then depth(/u1D434) = 4,...

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Under this assumption, both parts reduce to the following

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc1.2 We ﬁrst prove /T_heorem1.2 in the special case, where /u1D43F=/u1D458/u1D45F. Under this assumption, both parts reduce to the following. Proposition 3.1. Let/u1D43Abe an algebraic /u1D458-group and let /u1D434⊂/u1D43Abe a ﬁnite abelian /u1D458-subgroup of rank/u1D45F. Assume that char(/u1...

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As we pointed out in the Introduction, we have (4.1) /u1D4613(/u1D43A)| /u1D4612(/u1D43A)| /u1D461(/u1D43A)

V /a.sc/r.sc/i.sc/a.sc/t.sc/i.sc/o.sc/n.sc/s.sc /o.sc/n.sc /t.sc/h.sc/e.sc /t.sc/o.sc/r.sc/s.sc/i.sc/o.sc/n.sc /i.sc/n.sc/d.sc/e.sc/x.sc In (1.1)-(1.3), we deﬁned the torsion indices /u1D461(/u1D43A),/u1D4612(/u1D43A) and/u1D4613(/u1D43A) for any aﬃne/u1D458-group/u1D43A. As we pointed out in the Introduction, we have (4.1) /u1D4613(/u1D43A)| /u1D4612(/u1...

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/T_his notion was introduced by Serre in [ Ser58] (reprinted in [ Ser01]), where he also showed that special /u1D458-groups are smooth, aﬃne and connected

P/r.sc/o.sc/o.sc/f.sc/s.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc/s.sc1.1 /a.sc/n.sc/d.sc1.5 Recall that an algebraic /u1D458-group/u1D43Ais called special if/u1D43B1(/u1D43E,/u1D43A) = 1 for every ﬁeld extension /u1D43E//u1D458. /T_his notion was introduced by Serre in [ Ser58] (reprinted in [ Ser01]), where he also showed that special /u1D458-groups...

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Lemma 6.1

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc1.3 Our proof is based on the following lemma. Lemma 6.1. Let/u1D43Abe a semisimple/u1D458-group. For any proper parabolic subgroup/u1D443⊂/u1D43A, let/u1D43F/u1D443⊂/u1D443 denote a Levi subgroup. /T_he index/u1D4613(/u1D43A) is the least common multiple of the depths of all maximal ﬁnite ab...

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(See [ Tot05a] for torsion indices of /u1D438sc 6 and/u1D438sc 7 .) We conclude that ( 6.4) holds in each of the cases (i) - (viii), as we wanted to show

= 12, respectively. (See [ Tot05a] for torsion indices of /u1D438sc 6 and/u1D438sc 7 .) We conclude that ( 6.4) holds in each of the cases (i) - (viii), as we wanted to show. □ Remark 6.3. Let/u1D43Abe an aﬃne group over an algebraically closed ﬁeld /u1D458. Recall that /u1D461(/u1D43A) is deﬁned as the least common multiple of ind(/u1D44B), as/u1D44Brang...

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Let/u1D458be a ﬁeld of characteristic zero, and let /u1D43Abe a semisimple/u1D458-group of type /u1D4388

D/e.sc/p.sc/t.sc/h.sc/s.sc /o.sc/f.sc /t.sc/h.sc/e.sc /m.sc/a.sc/x.sc/i.sc/m.sc/a.sc/l.sc /f.sc/i.sc/n.sc/i.sc/t.sc/e.sc /a.sc/b.sc/e.sc/l.sc/i.sc/a.sc/n.sc /s.sc/u.sc/b.sc/g.sc/r.sc/o.sc/u.sc/p.sc/s.sc /o.sc/f.sc/u1D4388 In this section we prove Proposition 6.2. Let/u1D458be a ﬁeld of characteristic zero, and let /u1D43Abe a semisimple/u1D458-group of ty...

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We denote the /u1D456-th coordinate of a word /u1D464∈/u1D436by/u1D464/u1D456

Elements of /u1D436are called words. We denote the /u1D456-th coordinate of a word /u1D464∈/u1D436by/u1D464/u1D456. /T_heHamming weight of/u1D464∈ F/u1D45B 2 is the number of non-zero coordinates wt(/u1D464) = /barex /barex{1≤/u1D456≤/u1D45B:/u1D464/u1D456= 1} /barex /barex. We use the notation (F/u1D45B 2)0 ⊂ F/u1D45B 2 for the subspace consisting of all...

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