Abelian varieties genuinely of $\mathrm{GL}_n$-type

Enric Florit; Francesc Fit\'e; Xavier Guitart

Abelian varieties genuinely of $\mathrm{GL}_n$-type

T0 review · 0 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that an abelian variety genuinely of $\mathrm{GL}_n$-type and geometrically of the first kind carries $\lambda$-adic Galois representations that are symplectic or orthogonal according to the building block's Albert type.

desk verdict Solid generalization of Ribet and Chi-BGK with a real explicit family; the main theorem holds up, but Prop 6.10 has a proof gap that should be fixed or removed before publication. read the letter →

arxiv 2412.21183 v3 pith:3GTCZEG5 submitted 2024-12-30 math.NT

Francesc Fité , Enric Florit , Xavier Guitart This is my paper · ORCID

classification math.NT MSC 11G1011F8014K02

keywords abelianvarietiesofGL_n-typegenuinelybuildingblocksinnertwistsnebentypecharacterssymplecticGaloisrepresentationsorthogonalJacobiansgenus2curves

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 develops a theory of abelian varieties genuinely of $\mathrm{GL}_n$-type, meaning simple varieties whose $n$-dimensional Galois representation is not explained by smaller factors. Its main theorem states that if the building block $B$—the simple isogeny factor of $A_{\bar k}$—is of the first kind, i.e. the center of $\mathrm{End}^0(B)$ is totally real, then the $\lambda$-adic Galois representations attached to $A$ are still symplectic or orthogonal, just as they were under the older, stronger hypothesis that the center of $\mathrm{End}^0(A)$ itself is totally real. The paper constructs an absolutely irreducible $H_\lambda$-vector space $W_\lambda(A)$ of dimension $n$, a nebentype character $\varepsilon$, and a non-degenerate Galois-equivariant pairing $W_\lambda(A)\times W_\lambda(A)\to H_\lambda(\varepsilon\chi_\ell)$ whose alternating or symmetric nature is dictated by the Albert type of $B$. It closes by building an explicit family of genus 2 curves over a quadratic field whose restrictions of scalars are fourfolds genuinely of $\mathrm{GL}_4$-type, including one with center $\mathbb{Q}(\sqrt{-2})$ where the older theorems do not apply.

What carries the argument

The argument turns on four objects: the building block $B$, the absolutely simple isogeny factor of $A_{\bar k}$, with $\mathrm{End}^0(B)$ a central division algebra over a totally real field $F$; the inner twists $\chi_\gamma(s)=\gamma\alpha(s)/\alpha(s)$ and their nebentype $\varepsilon=\chi_c^{-1}$, which measure how the Galois group of the minimal field $K$ where all endomorphisms are defined acts on $H$; the crossed-product descent criterion that realizes $\varrho_L$ over $H_\lambda$ exactly when $\lambda\in\Sigma_A$; and the cited theorem for building blocks, which provides the $F_l$-bilinear, non-degenerate, $G_K$-equivariant pairing on $W_l(B)$. The main work is to show that $W_\lambda(A)$ and $W_l(B)\otimes_{F_l}H_\lambda$ are isomorphic as $G_K$-modules and then to use Schur's lemma, twisted by $\varepsilon$, to promote that pairing to $G_k$-equivariance.

What would settle it

Compute the Frobenius polynomials of the compatible system for the explicit fourfold $A_\alpha$ of Example 8.6 at a good prime $\mathfrak p$ outside the bad set. The theorem requires the multiset of eigenvalues of $\varrho_\lambda(\mathrm{Fr}_{\mathfrak p})$ to be stable under $\alpha\mapsto\varepsilon(\mathrm{Fr}_{\mathfrak p})\mathrm{Nm}(\mathfrak p)/\alpha$; a single prime violating this self-twist relation would show the claimed $G_k$-equivariant pairing cannot exist.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is the theorem in Section 7: if $A$ is genuinely of $\mathrm{GL}_n$-type and geometrically of the first kind, then for every prime $\lambda$ of $H=Z(\mathrm{End}^0(A))$ lying over a splitting prime of $\mathrm{End}^0(B)$ and satisfying $\lambda\in\Sigma_A$, there is an absolutely irreducible $H_\lambda[G_k]$-module $W_\lambda(A)$ of dimension $n$ and an $H_\lambda$-bilinear, non-degenerate, $G_k$-equivariant pairing $$\psi_\$\lambda$: W_\$\lambda$(A)\times W_\$\lambda$(A)\to H_\$\lambda$(\varepsilon\chi_\ell),$$ alternating if $B$ has Albert type I or II and symmetric if $B$ has type III. The representation $\varrho_\lambda$ therefore lands in the general symplectic group in the first two cases and in the general orthogonal group in the third, with similitude character $\varepsilon\chi_\ell$. This is exactly the symplectic/orthogonal conclusion of the earlier results, achieved with the totally-real condition placed on $F=Z(\mathrm{End}^0(A_{\bar k}))$ rather than on $H$.

Load-bearing premise

The load-bearing premise is that the building block $B$ has one of the three non-CM Albert types I, II, or III and that the cited theorem supplies a non-degenerate Galois-equivariant pairing on its $l$-adic module, for without that pairing the descent step has nothing to transfer.

Editorial extensions

If this is right

For any $A$ in the theorem, the compatible system $\{\varrho_\lambda\}$ is a family of symplectic or orthogonal representations, so its Frobenius polynomials satisfy the self-reciprocal-up-to-nebentype relation in which each eigenvalue $\alpha$ is paired with $\varepsilon(\mathrm{Fr}_{\mathfrak p})\mathrm{Nm}(\mathfrak p)/\alpha$.
The determinant formula $\delta_\lambda=\varepsilon^{n/2}\chi_\ell^{n/2}$ fixes the scalar twists of these representations, and when $H$ is totally real the nebentype disappears and $\delta_\lambda=\chi_\ell^{n/2}$.
The paper's genus-2 construction yields abelian fourfolds genuinely of $\mathrm{GL}_4$-type with $\mathrm{End}^0(A_\alpha)\simeq\mathbb{Q}(\sqrt{-2})$ or $\mathbb{Q}(\sqrt{2})$; the $\mathbb{Q}(\sqrt{-2})$ example is a case where the previous symplectic/orthogonal theorems did not apply but the new theorem does.
The criterion in Section 9 lets one certify $\mathrm{End}(A_{\bar k})=\mathbb{Z}$ from two ordinary, stably irreducible, good-reduction primes whose Frobenius fields are linearly disjoint, which is how the concrete fourfold examples are verified.

Reading between the lines

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

The splitting condition $\lambda\in\Sigma_A$ is not merely technical: Propositions 6.2 and 6.4 show it is exactly the obstruction to descending $\varrho_L$ to $H_\lambda$, so one could ask what invariants or generalized pairings appear when the endomorphism algebra is nonsplit at $\lambda$.
The same descent-plus-nebentype strategy should apply to other settings where a Galois representation is realized over a field only after twisting by a finite-order character, with $\varepsilon$ serving as a template for twisted self-duality beyond the classical symplectic/orthogonal dichotomy.
The Richelot-isogeny family in Section 8 suggests a search for higher-dimensional analogues: Jacobians of higher-genus curves over quadratic fields that are isogenous to their Galois conjugates should produce genuinely $\mathrm{GL}_{2g}$-type varieties, and the stable-irreducibility criterion offers a practical certification route.

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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

Short version: the main theorem is real and the proof route checks out; the only concrete gap I can find is Proposition 6.10, a side result not used in Section 7. I would send this to a serious referee rather than desk reject.

What is new: the definition of genuinely GL_n-type, the building-block/inner-twist/nebentype package, and the descent theorem that replaces "H totally real" with "F totally real" in the Chi-Banaszak-Gajda-Krasoń symplectic/orthogonal image theorem. That is a genuine extension, not just notation. The construction in Sections 8-9 of explicit abelian fourfolds genuinely of GL_4-type with nontrivial nebentype is concrete and checks out as far as I can tell; the Richelot isogeny computation in Proposition 8.1 is a nice piece of work.

The central proof is sound in structure. Proposition 2.9 gives a clean characterization; Section 6's realizability over H_lambda is standard descent via a 2-cocycle, and the vanishing of the class at split primes is correct. The descent of the BGK pairing in Section 7 is the right argument: absolute irreducibility plus Schur's lemma forces the twist to be the nebentype, and Lemma 6.7 rules out ambiguity. Reliance on Theorem 7.2 is legitimate use of a known external theorem; the "geometrically first kind" hypothesis supplies exactly the Albert types and split primes needed.

The one soft spot is Prop 6.10. The wedge argument does not control the sign of the square root when passing from the isomorphism to the determinant identity, and as written the proof is not valid. The stress-test note is right that this result is not used in the proof of Theorem 7.5, so the main theorem is not threatened. But it should either be repaired or explicitly removed or flagged as unproved, since it appears as a proposition.

Minor: I did not verify every trace computation in Section 6, but nothing there looks off. The paper's reliance on [Gui10], [Pyl04], and [Rib92] is appropriate; self-citations are not problematic here.

Who this is for: arithmetic geometers working on Galois representations attached to abelian varieties, and anyone who wants the GL_2/Pyle theory in higher dimension. The reader gets a working framework and a family of examples, not just a theorem statement.

Recommendation: conditional accept at an appropriate journal. The main theorem deserves referee time; the referee should ask the authors to fix or remove Prop 6.10. I would accept for peer review.

Referee Report

0 major / 4 minor

Summary. The paper introduces abelian varieties "genuinely of GL_n-type" (simple, of GL_n-type, whose base change has no simple factor of smaller GL_m-type), generalizing Ribet's GL_2-type. It develops a theory of building blocks, inner twists, and nebentypes for such varieties, and constructs, for primes where the endomorphism algebra splits, an H_λ-vector space W_λ(A) of dimension n with an absolutely irreducible G_k-representation ϱ_λ. The main theorem states that, under the weaker condition that the center F of End^0(A_{k̄}) is totally real (geometrically of the first kind), there exists a non-degenerate G_k-equivariant pairing W_λ(A) × W_λ(A) → H_λ(εχ_ℓ), alternating for Albert types I/II and symmetric for type III. This extends results of Chi, Banaszak, Gajda, and Krasoń, which required the center H of End^0(A) to be totally real. The paper also constructs an explicit family of abelian fourfolds genuinely of GL_4-type, obtained as restrictions of scalars of genus-2 Jacobians over quadratic fields, with a criterion for trivial endomorphism ring.

Significance. If correct, this is a meaningful extension of the known symplectic/orthogonal image theorem, replacing the hypothesis that the endomorphism algebra center is totally real by the weaker and more natural condition that only the geometric center is totally real. The paper gives a self-contained treatment of building blocks, inner twists, and nebentypes, with detailed proofs of the descent to H_λ and the G_k-equivariance via Schur's lemma. The explicit family in §8 and the endomorphism-triviality criterion in §9 are concrete and checkable, providing a useful testbed. The reliance on the cited BGK theorem (Theorem 7.2) is standard and the hypotheses match. The main derivation appears sound; the only concrete mathematical gap I found is in Proposition 6.10, which is not used in the proof of the main theorem.

minor comments (4)

[§6, Proposition 6.10] The proof is not valid as written. From (6.8), taking determinants only yields δ_λ^2 = ε^n χ_ℓ^n; the argument using ∧^{n/2} does not control the sign of the square root. The assertion δ_λ = ε^{n/2}χ_ℓ^{n/2} needs an additional argument, for example using the Rosati involution or the polarization, which is not supplied. Since Proposition 6.10 is not used in §7, this does not affect the main theorem, but the statement should be proved correctly or removed.
[§5, proof of Proposition 5.3] The map α is introduced as α: ∆ × ∆ → E^×, but the formula c_V(g,h) = α(g)·gα(h)·α(gh)^{-1} indicates that α should be a 1-cochain α: ∆ → E^×. This is a typo in the domain of α.
[§1 and §5] There are minor typos: in §1, "the the 2-cohomology class" has a duplicated article; in §5, "the later condition" should be "the latter condition." The transliteration "Chebotaryov" is nonstandard; consider "Chebotarev."
[Introduction, Theorem] The phrasing of part ii) may suggest that the condition λ lying over a split prime of End^0(B) is a hypothesis for the existence of ψ_λ. In fact, Proposition 6.9 constructs ψ_λ for every λ ∈ Σ_A, and the extra condition on l ∈ Σ_B is needed only to determine the alternating or symmetric type. Clarifying this would help the reader.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the nebentype twist is derived from inner twists, and the pairing is imported from the external BGK theorem; self-citations are background only.

full rationale

Walking the claimed derivation chain: the main theorem's pairing is produced in Proposition 6.9 and Theorem 7.5. The nebentype ε is not a fitted parameter: Definition 4.4 sets ε = χ_c^{-1} from the inner twist χ_c(s)=cα(s)/α(s), and Proposition 6.6 proves W_λ(A) ≅ ε ⊗ W_λbar(A) from the conjugation formula (3.1) and the equality (6.5); the same ε appears in (6.8)–(6.9) by Schur's lemma and the uniqueness Lemma 6.7. The G_K-equivariant pairing is imported from the external theorem of Chi, Banaszak, Gajda and Krasoń (Theorem 7.2), whose Albert-type hypotheses are exactly supplied by the 'geometrically of the first kind' assumption. Proposition 7.4 identifies W_λ(A)|_{G_K} with W_l(B) ⊗ H_λ using Faltings' theorem and Frobenius trace comparisons; Theorem 7.5 then proves that the G_k-character of the pairing is ε, not by assumption. The eigenvalue relation α = χ_ℓ(Fr_p)/α used in Proposition 6.9 is the standard Weil-pairing consequence, not the conclusion being proved. The cited works of the present authors ([Gui10], [Gui12], [Fit24], [GQ14]) supply background, converse, or example machinery; none of them provides the load-bearing symplectic/orthogonal statement. The only concrete defect I find, the unproved sign in Proposition 6.10's δ_λ = ε^{n/2}χ_ℓ^{n/2}, is not used in the proof of Theorem 7.5; it is a side-result correctness concern, not a circularity. Overall, no step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central theorem is a theorem about existing objects, not a postulate of new entities. It relies on deep prior results (Faltings, Tate, Albert, Chebotaryov, Chi-BGK, Guitart) and on the stated geometric first-kind hypothesis. There are no free parameters fitted to data.

assumptions (9)

standard math Albert's classification of division algebras with positive involution (Albert types I-IV) and dimension constraints
Used in Example 2.5, Proposition 3.10 and Remark 3.9 to relate Albert types of A and its building block B.
standard math Faltings' isogeny theorem and the identification End_{Qℓ[G_L]}(Vℓ(A)) ≃ End^0(A_L) ⊗ Qℓ (Fal83, Satz 4)
Used in Lemma 6.1, Proposition 6.2, Lemma 6.5 to compute endomorphism algebras from Galois invariants of Tate modules.
standard math Chebotaryov density theorem
Used in Lemma 6.3, Proposition 7.4 and elsewhere to pass from Frobenius trace equalities to isomorphisms of semisimple representations.
standard math Skolem-Noether theorem
Used in Proposition 3.4 to find α(s) realizing the Galois action on End^0(A_kbar) by inner automorphisms.
standard math Tate's theorem on endomorphisms of abelian varieties over finite fields (Tat66)
Used in the proof of Proposition 9.4 to identify End^0(A_{p,n}) with the field K_{p,n} for ordinary stably irreducible reduction.
domain assumption Chi-Banaszak-Gajda-Krasoń theorem attaching a symplectic/orthogonal pairing to building blocks of Albert type I, II, III (Theorem 7.2)
Black-box used in Section 7 to obtain the initial pairing ψ_l on W_l(B) over the field K where all endomorphisms are defined.
domain assumption Guitart's existence theorem for abelian varieties with prescribed absolutely simple factors (Gui12, Theorem 2.5)
Used in Proposition 3.5 to prove the converse direction from building blocks to varieties genuinely of GL_n-type.
domain assumption In Section 8, parameters are assumed to give a genus 2 curve with End(Jac(C)_Qbar) = Z; the example verifies this via the criterion in Proposition 9.4
This is a stated genericity hypothesis for the family of fourfolds; the paper checks it for the explicit Example 8.6 using primes 5 and 11.
standard math Richelot isogeny construction and Smith's theorem identifying the transpose correspondence with the Rosati dual (CF96, Smi05)
Used in §8 to compute µ_s and the cocycle cα(s,s)=±2.

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Cite this review

Pith. "Pith review of Abelian varieties genuinely of $\mathrm{GL}_n$-type." pith.science (2026). https://pith.science/paper/3GTCZEG5

@misc{pith2026241221183,
  author       = {Pith},
  title        = {Pith review of: Abelian varieties genuinely of $\mathrmGL_n$-type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3GTCZEG5}},
  note         = {Machine review of arXiv:2412.21183}
}

abstract

A simple abelian variety $A$ defined over a number field $k$ is called of $\mathrm{GL}_n$-type if there exists a number field of degree $2\dim(A)/n$ which is a subalgebra of $\mathrm{End}^0(A)$. We say that $A$ is genuinely of $\mathrm{GL}_n$-type if its base change $A_{\overline{k}}$ contains no isogeny factor of $\mathrm{GL}_m$-type for $m<n$. This generalizes the classical notion of abelian variety of $\mathrm{GL}_2$-type without potential complex multiplication introduced by Ribet. We develop a theory of building blocks, inner twists and nebentypes for these varieties. When the center of $\mathrm{End}^0(A)$ is totally real, Chi, Banaszak, Gajda, and Kraso\'n have attached to $A$ a compatible system of Galois representations of degree $n$ which is either symplectic or orthogonal. We extend their results under the weaker assumption that the center of $\mathrm{End}^0(A_{\overline{k}})$ be totally real. We conclude the article by showing an explicit family of abelian fourfolds genuinely of $\mathrm{GL}_4$-type. This involves the construction of a family of genus 2 curves defined over a quadratic field whose Jacobian has trivial endomorphism ring and is isogenous to its Galois conjugate.

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Banaszak, W

G. Banaszak, W. Gajda, and P. Krasoń. On the image of -adic Galois representations for abelian varieties of type I and II. Doc. Math., Extra Vol. (2006), 35–75

work page 2006

[2] [2]

Banaszak, W

G. Banaszak, W. Gajda, and P. Krasoń. On the image of Galois l-adic representations for abelian varieties of type III. Tohoku Math. J. (2) 62 (2) 163--189 (2010)

work page 2010

[3] [3]

Banaszak, A

G. Banaszak, A. Kaim-Garnek. The Tate module of a simple abelian variety of type IV . New York J. Math. 27, 1240--1257 (2021)

work page 2021

[4] [4]

Bourbaki

N. Bourbaki. Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples. Second revised edition of the 1958 edition, Springer, Berlin, 2012. x+489 pp

work page 1958

[5] [5]

Cassels, E.V

J.W.S. Cassels, E.V. Flynn. Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2 LMS Lecture Note Series, vol. 230, Cambridge University Press, 1996

work page 1996

[6] [6]

Cantoral-Farf \'a n, D

V. Cantoral-Farf \'a n, D. Lombardo, and J. Voight. Monodromy groups of Jacobians with definite quaternionic multiplication. Preprint arXiv:2303.00804

arXiv

[7] [7]

W. Chi. Twists of Central Simple Algebras and Endomorphism Algebras of Some Abelian Varieties, Math, Ann. 276, 615--632 (1987)

work page 1987

[8] [8]

W. Chi. On the Tate module of absolutely simple abelian varieties of type II, Bulletin of the Institute of Mathematics Academia Sinica Volume 18, Number 2 (1990)

work page 1990