Automorphism groups of curves with simple Jacobians

Luca Mauri

REVIEW 1 major objections 5 minor 15 references

Automorphism groups of curves with simple Jacobians

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Pith's one-line read A curve whose Jacobian is simple has automorphism group cyclic, quaternion, or trivial — and every such group occurs.

desk verdict A serious classification paper that likely has the right shape, but the quaternion existence half rests on omitted degeneration proofs — send it to review with a request to expand Section 6. read the letter →

arxiv 2607.20397 v1 pith:JP5OSJ2G submitted 2026-07-22 math.AG math.NT

Luca Mauri This is my paper · ORCID

classification math.AGmath.NT MSC 14H4014K1214H3714K0514H45

keywords simpleJacobianautomorphismgroupofacurvesuperellipticcurvesfixed-point-freerepresentationsendomorphismalgebrasspecializationcomplexmultiplicationquaternion

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 settles the classification of automorphism groups of smooth projective curves of genus at least two over an algebraically closed field of characteristic zero whose Jacobian is simple (i.e., not isogenous to a product of lower-dimensional abelian varieties). The answer is short: the group must be cyclic of prime-power order, cyclic of order pq for distinct primes with pq ≠ 6, a generalized quaternion group, or trivial — and every group on this list actually occurs. The classification half is quick: simplicity forces every nontrivial automorphism to have a rational quotient, so the action on holomorphic differentials is fixed-point-free and the classification of finite groups with such representations applies. The existence half is the hard part: explicit superelliptic families are constructed, and a new specialization argument shows that 100% of the members (in a height-density sense) have simple Jacobian with the prescribed endomorphism ring. If correct, the results complete the answer to the question of which finite groups can act on a curve without splitting its Jacobian, and they produce many new examples of curves whose Jacobians carry unusually large endomorphism algebras.

What carries the argument

The load-bearing tool is a specialization lemma: for an admissible degeneration of a family of curves to a reducible fiber, the geometric endomorphism algebra of the generic Jacobian embeds into that of the special fiber, preserving dimensions of idempotent kernels, and torsion subgroups specialize bijectively. This enables an induction on the number of branch points: degenerate to a union of lower-genus curves, compare isotypic components, and rule out unexpected factors. The base cases are handled by complex multiplication theory, reading the CM type off the explicit equation; for the quaternion family, the endomorphism algebra is shown to be a definite quaternion division algebra. The cla

What would settle it

Compute the endomorphism algebra of the Jacobian of a concrete small member of the predicted families, e.g., C: y^3 = x(x^8 − 2)(x^8 − 3) over Q (p=3, n=2, L=2); the theorem predicts End^0(Jac(C)) ≅ Q(ζ_9). Using reduction modulo a prime ℓ ≠ 3 and counting points, or a period computation, any indication of an endomorphism not in Q(ζ_9), or a decomposition into lower-dimensional factors, would falsify the density-one claim for that family.

Watch

Extended reading notes

Core claim

Theorem 1.2: if C is a curve of genus ≥ 2 over an algebraically closed field of characteristic 0 with simple Jacobian, then Aut(C) is isomorphic to C_{p^n} for some prime p and n ≥ 1, or C_{pq} for distinct primes with pq ≠ 6, or the generalized quaternion group G_{2^n}, or the trivial group. Conversely, for every group in this list, there exists such a curve. The proof decouples into a short classification (simplicity implies fixed-point-free action on regular differentials, the classification of finite groups admitting such representations, plus superelliptic genus constraints) and a long existence argument that constructs families of curves y^p = x·∏(x^{p^{n-1}} - a_l) for the cyclic case

Load-bearing premise

The existence proofs rest on the specialization lemma, which assumes the chosen families admit semistable regular models over A^1 with Néron models whose special fibers are abelian varieties, and that endomorphism algebras and torsion subgroups behave faithfully under specialization; if any constructed degeneration violates these assumptions, the density-one simplicity conclusions do not follow.

Editorial extensions

If this is right

Every group in the classification is realized by a family of curves over Q; for 100% of the parameter values (ordered by height) the Jacobian is geometrically simple and its endomorphism ring is explicitly determined (Z[ζ_{p^n}] for cyclic p^n, and a definite quaternion order for G_{2^n}).
The curve y^p = x^q − 1 is the only curve over Q (up to isomorphism) with automorphism group C_{pq}, pq ≠ 6, and simple Jacobian; no such curve exists for pq = 6.
The constructed families give infinitely many new examples of unlikely intersections in moduli spaces of principally polarized abelian varieties: for genus g ≥ 9, the locus of Jacobians whose endomorphism ring contains Z[ζ_{p^n}] has negative expected dimension, yet the families land in it.
When the family has at least two branch-point parameters (L ≥ 2), the curves realize infinitely many distinct isomorphism classes and their Jacobians infinitely many distinct isogeny classes over Q.
The results pin down the possible automorphism groups of curves with simple Jacobians, resolving the motivating question completely; in particular the only non-abelian groups that can occur are the generalized quaternion groups G_{2^n}.

Reading between the lines

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

The specialization lemma likely has broader use: it converts a generic simplicity statement into a check on degenerate fibers, and the torsion bijection is a new tool for controlling torsion in families of abelian varieties; one could apply it to other families of superelliptic curves or to Prym varieties.
The connection to the classification of fixed-point-free representations suggests a geometric principle: curves with simple Jacobians are precisely those whose automorphism groups are space-form groups, the same groups that act freely on spheres. For higher-dimensional varieties or for curves over non-closed fields, analogous constraints might follow from a fixed-point-free condition on differenti
The density-one statement is measured by height and leaves open whether the exceptional set is finite or Zariski-closed; testing small parameter values (e.g., the families with L=2) could reveal whether the simplicity condition holds for all but finitely many fibers, a stronger statement than the paper claims.
The p=2, n>2 case shows a precise role of CM types: when the generic CM type has nontrivial stabilizer, the generic Jacobian is not simple, which is exactly why the theorem requires L≥3 in that case; this predicts that any family improving the L-bound would have to break the CM-stabilizer symmetry.

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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

The headline: this is a serious paper that likely has the right general shape, but the quaternion existence half is under-supported. The reader's conditional verdict is about right.

What is actually new: a complete classification answer to Question 1.1, filling gaps left by Zarhin, CLV23, and the Catalan curve literature. The classification half (Theorem 3.1) is clean: fixed-point-free representation, Wolf's list, superelliptic genus restrictions. The cyclic prime-power constructions in Section 4 are detailed, with CM-type arguments for L=1 and specialization/induction for larger L. The uniqueness of the C_pq example (Theorem 5.1) is a genuine addition. The paper also gives density-one families and notes the unlikely intersections angle.

The soft spot is exactly where the reader and the stress-test point: Section 6. Theorem 6.1, and hence Theorem 1.2(iii), relies on Proposition 6.13 (an admissible degeneration to Jac(C1) x Jac(C2)^2) and Corollary 6.14. The paper says the proofs are 'conceptually identical' or 'similar' to Section 4, but the computations involve a multistep blow-up and explicit 2-torsion specialization. In Section 4 the analogous Proposition 4.16 is actually proved in detail; Section 6 is not. Since CLV23 only handled n=1, all new quaternion cases depend on these omitted details. If the rational form of the second component (twist vs. C2) or the torsion specialization is wrong, the induction and the existence claim break. This is a real gap in support, not a stylistic quibble.

The paper is open about the omission, which is honest, but a main theorem should not rest on that. The remainder of the paper is, as far as I can tell, sound: the classification chain is coherent, and the density-one machinery (Masser-style specialization, Lemma 2.21) is standard.

Who should read it: anyone working on automorphism groups of curves, simple Jacobians, or CM constructions in moduli. It deserves a serious referee. I would send it to review and ask the author to write out Section 6.3 (or at least provide the blow-up computations and the torsion specialization verification) before acceptance.

Recommendation: engage; conditional accept after expansion of Section 6.

Referee Report

1 major / 5 minor

Summary. The paper determines, for curves of genus at least 2 over an algebraically closed field of characteristic 0 with simple Jacobian, all possible automorphism groups, and proves that every group in the list occurs. The classification part proceeds by showing that the action of Aut(C) on H^0(C,Ω^1) is fixed-point-free, then applying Wolf's classification together with superelliptic genus restrictions; the resulting list is cyclic p-power, cyclic product of two distinct primes except 6, generalized quaternion, or trivial. The constructive part is substantial: for cyclic p-power groups the author gives explicit families y^p = x∏(x^{p^{n-1}}-a_l), proves generic endomorphism algebra Q(ζ_{p^n}) by a combination of CM-type arguments, specialization techniques, and induction on the number of branch points, and obtains density-one simplicity and automorphism-group statements; for pq-cyclic groups the Catalan curve is shown to be the unique example; for generalized quaternion groups the author constructs hyperelliptic families y^2 = x(x^{2^{n+1}}-1)∏(x^{2^n}-a_l)(x^{2^n}-a_l^{-1}) and claims the analogous endomorphism-algebra computation. The paper also derives density-one results using height-counting and Masser's specialization bound.

Significance. If correct, this is a complete answer to a natural and long-studied question, and it is significantly stronger than prior results, which were largely limited to cyclic prime order or the classical quaternion group of order 8. The modular-unlikely-intersection consequences are also new. The paper is honest about what is proved and what is deferred: the cyclic constructions are written out in detail, including the delicate specialization and torsion-basis computations, and the author explicitly flags the parts of the quaternion case that are left as analogues of earlier arguments. This is a high-value contribution to the arithmetic geometry of Jacobians.

major comments (1)

[Section 6.3, Prop. 6.13, Cor. 6.14, Thm. 6.12] The new quaternion construction for n≥2 is not proved at the points where it is load-bearing. The section preamble states that many proofs are omitted; Proposition 6.13 is justified only as 'conceptually similar' to [CLV23, Lemma 6.3.2] and Proposition 4.16, Corollary 6.14 says its proof is 'identical' to Corollary 4.18, and Theorem 6.12 says its proof is 'identical' to Theorem 4.10. These are not cosmetic omissions. Proposition 6.13 supplies the admissible degeneration J_{T',t} ≅ Jac(C1)×Jac(C2)^2 and, crucially, the explicit specialization of the 2-torsion basis (6.3), including the new points P_i and the asymmetric assignment of D^±_{α_l,i} to the two copies of Jac(C2). Corollary 6.14 uses those formulas to rule out nontrivial isotypic decompositions of J^al, and Theorem 6.1 relies on this to prove simplicity. Since [CLV23] treats only n=1, the entire existence part for G_{2^n} with n

minor comments (5)

[Section 6.3, formulas in Prop. 6.13(2)] The index ranges in the displayed formulas should be 0≤i≤2^{n+1}-1 and 0≤i≤2^n-1, not '1≤i≤2n+1' and '1≤i≤2n', to match the basis in (6.3).
[Theorem 2.5 and Section 6] The notation G_n in the hyperelliptic-automorphism table conflicts with the generalized quaternion group G_{2^n} used later. Please disambiguate, e.g. by renaming the table entries or the quaternion notation.
[Definition 1.4 / Definition 2.18] The height notation is introduced twice; in Definition 1.4 'H' is used before the formal definition, and the 'K L' in the text should be K^L. Minor but should be cleaned up.
[Section 5, proof of Theorem 5.1] The reduction 'every cyclic subgroup of order n of Aut(P^1) is conjugate to z↦ζ_n z' is only true over an algebraically closed field. Since automorphisms are geometric and uniqueness is over the algebraic closure, the argument is acceptable, but the text should say 'over an algebraic closure of Q' to avoid the false rational-statement.
[Remark 4.27] The discussion of [ACF25] and [Fit24] is interesting but unrelated to the main theorem; consider moving it to a separate note or removing it, as it distracts from the main thread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classification and construction chains are self-contained relative to external classical results and prior work, with no fitted parameter renamed as a prediction.

full rationale

The paper's classification half (Theorem 3.1) derives the list of possible automorphism groups from Lemma 2.1 and Theorem 2.2, which are consequences of the simplicity hypothesis, together with Wolf's classification of fixed-point-free representations and the standard classification of automorphism groups of hyperelliptic curves. These are external, non-circular inputs. The constructive half does not fit parameters to the desired conclusion: the families in Theorems 4.3 and 6.1 are explicit, the generic endomorphism algebra is computed from CM types and from the explicit action of automorphisms on differential forms, and the density-one specialization statements rest on the external specialization theory of Masser and of Cantoral-Farfán–Lombardo–Voight [CLV23]. The author is not an author of CLV23, so the reliance on CLV23 for Néron-model specialization lemmas is independent support rather than self-citation. The use of Theorem 3.1 inside Theorems 4.9 and 6.11 to identify Aut(C_a) is legitimate because the classification half is proved independently and does not assume the existence claims. The only significant weakness in the manuscript is that Section 6 explicitly says "we omit many of the details" for results parallel to Sections 4.3 and 4.4, and Proposition 6.13/Corollary 6.14 are asserted with proofs "similar" or "identical" to earlier statements. That is a verifiability/support gap, not a circular reduction: no claim is being justified by an unverified output of the same argument, and the missing computations are concrete degeneration statements whose analogues are already established for the cyclic case and for CLV23. Therefore no circular step meeting the quoted-reduction standard is present.

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

The central claim rests on established classification theorems and prior simplicity results; no ad hoc parameters or new postulated entities are introduced. The family parameters a_l are construction variables, not fitted constants. The most fragile external inputs are Masser's specialization theorem, the Néron-model lemmas adapted from CLV23, and the torsion-point linear-independence results of Wawrów.

assumptions (9)

standard math Wolf's classification of finite groups admitting irreducible fixed-point-free representations [Wol67, Theorems 6.1.11 and 6.3.1].
Used in Lemmas 3.12 and 3.13 to reduce the possible automorphism groups to a short list; this is an external completed classification.
standard math Bujalance–Gamboa–Gromadzki/Shaska classification of automorphism groups of hyperelliptic curves.
Theorem 2.5; used in Lemma 3.12 and throughout to constrain even-order automorphism groups.
standard math Feit–Thompson solvability of finite groups of odd order.
Invoked in Lemma 3.13 to apply Wolf's solvable-group classification.
standard math Masser's height-bound theorem for specializations of endomorphism rings.
Basis of Lemma 2.21 and hence of all '100% of points' statements; deep external input.
standard math Lang's CM theory: an abelian variety with a CM type with trivial stabilizer is simple with the prescribed endomorphism field.
Used in Lemmas 4.20, 4.22, and 4.25 via [Lan83, Theorem 3.5] to establish simplicity of low-L generic Jacobians.
standard math Zarhin's simplicity theorems for superelliptic Jacobians.
Theorem 4.1 supplies the base cases for the induction; cited from [Zar00; Zar26].
standard math Hazama/Goodson theorem that Catalan curves y^p = x^q - 1 have simple Jacobian and automorphism group C_{pq}.
The uniqueness theorem for C_{pq} (Theorem 5.1) uses this as the only existing example.
standard math Faltings isogeny theorem and Milne's injection of Aut(C) into finite-order units of End(J).
Used in Lemma 2.10 and Theorems 4.9/6.11 to translate endomorphism-algebra statements into automorphism-group statements.
standard math Torsion-point computations for superelliptic Jacobians [Waw21, §2].
Provides the F_p-bases J[1-Φ] and J[2] used in Corollaries 4.18 and 6.14.

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Pith. "Pith review of Automorphism groups of curves with simple Jacobians." pith.science (2026). https://pith.science/paper/JP5OSJ2G

@misc{pith2026260720397,
  author       = {Pith},
  title        = {Pith review of: Automorphism groups of curves with simple Jacobians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JP5OSJ2G}},
  note         = {Machine review of arXiv:2607.20397}
}

abstract

We classify all automorphism groups of smooth, projective, connected curves over an algebraically closed field of characteristic $0$ with simple Jacobians. For every nontrivial group in this classification, we construct a family of curves with that automorphism group and prove that a density-one set of members has simple Jacobian. These families yield infinitely many new examples of unlikely intersections in the moduli spaces of principally polarized abelian varieties.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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Geometrically simple counterexamples to a local-global principle for quadratic twists

[ACF25] Emiliano Ambrosi, Nirvana Coppola, and Francesc Fité. “Geometrically simple counterexamples to a local-global principle for quadratic twists”. In:Math. Z. 311.3 (2025), Paper No. 64, 15.url:https://doi.org/10.1007/s00209-025- 03858-9(cit. on p. 29). [Alb34a] A. Adrian Albert. “A solution of the principal problem in the theory of Riemann matrices”....

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Progr. Math. Birkhäuser, Boston, MA, 1991, pp. 177–192.url:https://doi.org/10.1007/978-1-4612- 0457-2_8(cit. on p. 3). [Lan+19] Aaron Landesman, Ashvin Swaminathan, James Tao, and Yujie Xu. “Surjectivity of Galois representations in rational families of abelian varieties”. In:Algebra Number Theory13.5 (2019). With an appendix by Davide Lombardo, pp. 995– ...

doi:10.1007/978-1-4612- 1991

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Specializations of endomorphism rings of abelian varieties

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Jacobians with complex multiplication

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Endomorphisms of superelliptic Jacobians

Progr. Math. Birkhäuser, Boston, MA, 2005, pp. 339–362.url:https://doi.org/10.1007/0- 8176- 4417-2_15(cit. on pp. 3, 15). 44 REFERENCES [Zar09] Yuri G. Zarhin. “Endomorphisms of superelliptic Jacobians”. In:Math. Z.261.3 (2009), pp. 691–707.url:https://doi.org/10.1007/s00209- 008- 0342- 5 (cit. on pp. 3, 15). [Zar18] Yuri G. Zarhin. “Endomorphism algebras...

doi:10.1007/0- 2005