Minimal Trivializing Isogenies of $\mathbb{G}_m$-gerbes over Abelian Varieties and Period-Index Problem

Ruoxi Li

A new invariant Ind_SH(α) defined from the geometry of Gm-gerbes proves the period-index conjecture for products of elliptic curves of any dimension.

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2026-06-29 00:12 UTC pith:OSYLPBIN

load-bearing objection The paper defines Ind_SH(α) via gerbe geometry and claims this settles the period-index conjecture for products of elliptic curves in any dimension. the 1 major comments →

arxiv 2605.30530 v1 pith:OSYLPBIN submitted 2026-05-28 math.AG

Minimal Trivializing Isogenies of mathbb{G}_m-gerbes over Abelian Varieties and Period-Index Problem

Ruoxi Li This is my paper

classification math.AG

keywords abelian varietiesBrauer groupperiod-index problemGm-gerbesisogeniessemi-homogeneous vector bundleselliptic curves

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces Ind_SH(α) as a refinement of the usual period-index relations for a Brauer class α on an abelian variety X. This invariant is extracted from the minimal trivializing isogenies of the associated Gm-gerbe by examining its μ_n-lifts and the 1-twisted semi-homogeneous vector bundles on the gerbe. The construction yields a concrete geometric description of how far α is from being trivialized by an isogeny. As a direct consequence the authors verify that the period-index conjecture holds for every product of elliptic curves, regardless of dimension.

Core claim

For an abelian variety X and α in its Brauer group, the invariant Ind_SH(α) is defined so that its value is completely determined by the minimal degree of an isogeny that trivializes the corresponding Gm-gerbe; this degree is recovered from the existence of suitable μ_n-lifts and from the properties of 1-twisted semi-homogeneous vector bundles on the gerbe. The resulting equality between Ind_SH(α) and the classical index establishes the period-index conjecture for all finite products of elliptic curves.

What carries the argument

Ind_SH(α), the invariant that assigns to each Brauer class α the minimal degree of a trivializing isogeny of its Gm-gerbe, obtained via μ_n-lifts and 1-twisted semi-homogeneous vector bundles.

Load-bearing premise

The proposed invariant Ind_SH(α) is well-defined on every Brauer class and its value is exactly the minimal isogeny degree needed to trivialize the gerbe.

What would settle it

An explicit computation, on a product of two or more elliptic curves, of a Brauer class whose minimal trivializing isogeny degree differs from the value predicted by Ind_SH(α) would falsify the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The period-index conjecture is true for every finite product of elliptic curves.
Ind_SH(α) supplies a geometric upper bound on the index that is at least as sharp as all previously known bounds.
Minimal trivializing isogenies of Gm-gerbes over abelian varieties can be read off from the existence of 1-twisted semi-homogeneous bundles.
The same method gives a uniform description of how the Brauer class behaves under isogenies of the base abelian variety.

Where Pith is reading between the lines

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

The construction may supply a practical algorithm for computing indices of Brauer classes on elliptic products by searching for the relevant twisted bundles.
If the invariant extends beyond elliptic products, it would give a uniform way to test the period-index conjecture on a larger class of abelian varieties.
The link between semi-homogeneous bundles and isogeny degrees suggests a possible dictionary between gerbe cohomology and moduli problems on the gerbe itself.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The paper defines Ind_SH(α) via gerbe geometry and claims this settles the period-index conjecture for products of elliptic curves in any dimension.

read the letter

The main takeaway is that this paper introduces the invariant Ind_SH(α) for a Brauer class α on an abelian variety and uses the geometry of the corresponding Gm-gerbe to prove that period equals index for products of elliptic curves in arbitrary dimension.

What is new is the invariant itself, built from μ_n-lifts and 1-twisted semi-homogeneous vector bundles, together with the study of minimal trivializing isogenies. The application extends prior results to this family of varieties in all dimensions.

The paper does a solid job picking a concrete open case of the period-index problem and framing it in terms of gerbe geometry rather than abstract Brauer group properties.

The soft spot is the complete absence of definitions, derivations, or proof sketches in the abstract. Without those, it is impossible to check whether Ind_SH(α) is independent of earlier quantities or whether the claimed equality actually follows from the construction. The central assumption that the invariant is well-defined and determined by the listed geometric data remains untested from the available information.

This work is aimed at specialists in Brauer groups and gerbes over abelian varieties. A reader already following the period-index literature on elliptic curves or abelian varieties could extract value from the geometric angle if the details hold up.

I would send it to peer review so that experts can examine the definitions and the application directly.

Referee Report

1 major / 0 minor

Summary. The paper proposes a new invariant Ind_SH(α) for a Brauer class α on an abelian variety X. The invariant is defined via the geometry of the associated G_m-gerbe, using μ_n-lifts and 1-twisted semi-homogeneous vector bundles on the gerbe. The authors apply this construction to prove that the period-index conjecture holds for products of elliptic curves of arbitrary dimension.

Significance. If the invariant is rigorously defined and the application to the period-index conjecture is correct, the result would advance understanding of Brauer groups and gerbes over abelian varieties by supplying a geometric refinement of period-index relations. A proof for products of elliptic curves would be a concrete step forward in a long-standing problem, though its broader impact depends on whether the construction extends or illuminates other cases.

major comments (1)

Abstract: The central claim that the period-index conjecture holds for products of elliptic curves rests on the well-definedness of Ind_SH(α) and its refinement of known relations, yet the abstract supplies neither an explicit definition nor a proof sketch; without these, the load-bearing step from the geometric data (μ_n-lifts and twisted bundles) to the conjecture cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to address their comments. We respond to the major comment below.

read point-by-point responses

Referee: Abstract: The central claim that the period-index conjecture holds for products of elliptic curves rests on the well-definedness of Ind_SH(α) and its refinement of known relations, yet the abstract supplies neither an explicit definition nor a proof sketch; without these, the load-bearing step from the geometric data (μ_n-lifts and twisted bundles) to the conjecture cannot be verified.

Authors: The abstract is intentionally concise and provides a high-level summary of the contributions, as is standard. The explicit definition of the invariant Ind_SH(α) appears in Definition 3.2, constructed via the minimal trivializing isogenies of the G_m-gerbe using its μ_n-lifts together with the existence of 1-twisted semi-homogeneous vector bundles. The refinement of period-index relations is developed in Section 4, and the proof that the period-index conjecture holds for products of elliptic curves of arbitrary dimension is given in Theorem 5.3. These sections contain the full details and arguments needed to verify the claims. We do not view the absence of technical definitions or sketches from the abstract itself as a deficiency, since abstracts are not intended to carry the full load of verification. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a new invariant Ind_SH(α) via the geometry of the Gm-gerbe, specifically through μ_n-lifts and 1-twisted semi-homogeneous vector bundles on the gerbe, then applies it to establish the period-index conjecture for products of elliptic curves. No quoted equations or steps reduce a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation chain; the central construction introduces independent content that refines existing relations without circular reduction to inputs. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the introduction of one new invariant without external verification and on standard facts about Brauer groups and gerbes.

axioms (1)

standard math Standard properties of the Brauer group Br(X) and Gm-gerbes over an abelian variety X
Used to define α ∈ Br(X) and the associated gerbe X

invented entities (1)

Ind_SH(α) no independent evidence
purpose: New invariant refining period-index relations
Introduced in the paper as a geometric refinement tied to minimal trivializing isogenies

pith-pipeline@v0.9.1-grok · 5637 in / 1142 out tokens · 26507 ms · 2026-06-29T00:12:51.354784+00:00 · methodology

0 comments

read the original abstract

For an abelian variety $X$ and $\alpha \in Br(X)$, we propose a new invariance $Ind_{SH}(\alpha)$ that refines the known period index relations. It is closely related to the geometry of $\mathcal{X}$, the $\mathbb{G}_m$-gerbe over $X$ that corresponds to $\alpha$: we study the minimal trivializing isogenies for $\mathcal{X}$ via its $\mu_n$-lifts and the $1-$twisted semi-homogeneous vector bundles on $\mathcal{X}$. As an application, we show that the period index conjecture holds true for products of elliptic curves of any dimension.

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

The prime divisors of the period and index of a Brauer class

[AW15] Benjamin Antieau and Ben Williams. “The prime divisors of the period and index of a Brauer class”. In:Journal of pure and applied algebra219.6 (2015), pp. 2218–2224. [EN83] G Elencwajg and MS Narasimhan. “Projective bundles on a complex torus.” In: (1983). [Hot25] James Hotchkiss. “The period-index problem for complex tori”. In:International Mathem...

work page arXiv 2015
[2]

Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties

[Li25] Ruoxi Li. “Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties”. In:arXiv preprint arXiv:2511.19282(2025). [Lie07] Max Lieblich. “Moduli of twisted sheaves”. In: (2007). [Muk78] Shigeru Mukai. “Semi-homogeneous vector bundles on an Abelian variety”. In:Journal of Math- ematics of Kyoto University18.2 (1978), pp...

work page arXiv 2025

[1] [1]

The prime divisors of the period and index of a Brauer class

[AW15] Benjamin Antieau and Ben Williams. “The prime divisors of the period and index of a Brauer class”. In:Journal of pure and applied algebra219.6 (2015), pp. 2218–2224. [EN83] G Elencwajg and MS Narasimhan. “Projective bundles on a complex torus.” In: (1983). [Hot25] James Hotchkiss. “The period-index problem for complex tori”. In:International Mathem...

work page arXiv 2015

[2] [2]

Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties

[Li25] Ruoxi Li. “Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties”. In:arXiv preprint arXiv:2511.19282(2025). [Lie07] Max Lieblich. “Moduli of twisted sheaves”. In: (2007). [Muk78] Shigeru Mukai. “Semi-homogeneous vector bundles on an Abelian variety”. In:Journal of Math- ematics of Kyoto University18.2 (1978), pp...

work page arXiv 2025