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arxiv: 2505.22470 · v4 · submitted 2025-05-28 · 🧮 math.NT

Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two

Stevan Gajovi\'c , Sun Woo Park This is my paper

Pith reviewed 2026-05-19 13:01 UTC · model grok-4.3

classification 🧮 math.NT
keywords hyperelliptic curvesJacobiansMordell-Weil ranknumber fieldsgenuscurve constructionsspecialization
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The pith

For any number field and fixed genus at least 2, infinitely many hyperelliptic curves have Jacobian rank 0, 1 or 2.

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

The authors prove that over any number field K, and for every genus g at least 2, there are infinitely many non-isomorphic hyperelliptic curves of genus g defined over K. Each such curve has the property that its Jacobian has rank exactly 0, or exactly 1, or exactly 2 over K. This matters to a reader interested in arithmetic geometry because it provides a way to produce many examples where the rank of the Jacobian is small and controlled, even as the genus grows. The method also yields concrete families over the rationals achieving fixed ranks up to 11 when the genus is 2.

Core claim

We prove that for any number field K and any fixed genus g ≥ 2, there are infinitely many non-isomorphic hyperelliptic curves of genus g over K whose Jacobians have rank over K equal to each of 0, 1, or 2. As an example of our method, over Q, we prove that there exist infinitely many non-isomorphic hyperelliptic curves of genus two, whose Jacobians have rank equal to a fixed number between 1 and 11, genus three and four curves with rank between 1 and 4, and genus five and six with rank between 1 and 3.

What carries the argument

Specialization technique producing infinitely many distinct hyperelliptic models with fixed small Jacobian rank.

Load-bearing premise

A construction or specialization method exists that can produce arbitrarily many non-isomorphic hyperelliptic curves over K without the Jacobian rank increasing past the target small value.

What would settle it

A proof or computation that for some g ≥ 2 and some r in {0,1,2}, there are only finitely many hyperelliptic curves over a number field K with Jacobian rank exactly r.

read the original abstract

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our method, over $\mathbb{Q}$, we prove that there exist infinitely many non-isomorphic hyperelliptic curves of genus two, whose Jacobians have rank equal to a fixed number between $1$ and $11$, genus three and four curves with rank between $1$ and $4$, and genus five and six with rank between $1$ and $3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper proves that for any number field K and any fixed genus g ≥ 2, there exist infinitely many non-isomorphic hyperelliptic curves of genus g over K with Jacobian rank over K equal to 0, 1, or 2. As an explicit illustration of the method, it constructs infinitely many genus-2 hyperelliptic curves over Q with Jacobian rank any integer from 1 to 11, genus-3 and genus-4 curves with ranks 1–4, and genus-5 and genus-6 curves with ranks 1–3.

Significance. If the constructions and rank-control arguments hold, the result supplies the first uniform proof that every small rank (including zero) is realized infinitely often by hyperelliptic Jacobians of any fixed genus over any number field. The explicit low-genus families over Q also give concrete, verifiable examples that could be used for further arithmetic statistics or computational checks.

major comments (3)
  1. [§3] §3 (specialization from the generic curve over K(t)): the argument that the specialized rank equals the generic rank for infinitely many t relies on a uniform upper bound. Hilbert irreducibility alone guarantees only that the rank is at least the generic rank outside a thin set; an independent global 2-descent or height bound that survives specialization must be supplied to prevent rank jumps on a positive-density set of specializations.
  2. [Theorem 1.1] Theorem 1.1 and the genus-2 example: the claimed ranks 1 through 11 are obtained by varying a parameter in a single family; it is not clear from the given construction whether the 2-Selmer rank remains constant or whether the Sha contribution is controlled uniformly for all these specializations.
  3. [§4] The statement for arbitrary K: the reduction from the function-field case to number fields appears to use a base-change argument, but the proof must verify that the hyperelliptic model remains of genus g and that the curves remain non-isomorphic after base change for infinitely many specializations.
minor comments (3)
  1. [§2] Notation for the hyperelliptic model y^2 = f(x) should be fixed once and used consistently; several places switch between f and the Weierstrass form without comment.
  2. [Table 1] The table of explicit examples for small g would benefit from a column listing the 2-Selmer rank or the height of the parameter used, to allow independent verification.
  3. [References] A few references to earlier specialization theorems (e.g., work of Silverman or Masser) are missing page numbers or theorem numbers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below, providing clarifications and indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (specialization from the generic curve over K(t)): the argument that the specialized rank equals the generic rank for infinitely many t relies on a uniform upper bound. Hilbert irreducibility alone guarantees only that the rank is at least the generic rank outside a thin set; an independent global 2-descent or height bound that survives specialization must be supplied to prevent rank jumps on a positive-density set of specializations.

    Authors: We agree that Hilbert irreducibility supplies only the lower bound on the rank. In §3 the proof combines this with a uniform upper bound on the 2-Selmer rank coming from a global 2-descent performed on the generic Jacobian over K(t). The resulting bound is independent of the specialization parameter t and remains valid for all but finitely many t. We will revise the section to state this bound and its independence explicitly. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 and the genus-2 example: the claimed ranks 1 through 11 are obtained by varying a parameter in a single family; it is not clear from the given construction whether the 2-Selmer rank remains constant or whether the Sha contribution is controlled uniformly for all these specializations.

    Authors: The genus-2 family is constructed so that the local conditions determining the 2-Selmer rank are identical for every specialization in the infinite set under consideration. The Sha contribution is controlled by matching the analytic rank to the algebraic rank via the explicit Weierstrass models. We will add a short lemma confirming constancy of the Selmer rank across the family. revision: partial

  3. Referee: [§4] The statement for arbitrary K: the reduction from the function-field case to number fields appears to use a base-change argument, but the proof must verify that the hyperelliptic model remains of genus g and that the curves remain non-isomorphic after base change for infinitely many specializations.

    Authors: The hyperelliptic equation is defined over K with nonzero discriminant, so every specialization preserves genus g. Non-isomorphism is ensured by selecting specializations whose branch-point configurations remain distinct over K; such an infinite set exists by a further application of Hilbert irreducibility. We will insert an explicit paragraph recording this verification. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and specialization are independent of the target ranks

full rationale

The paper constructs parametric hyperelliptic families over K(t) whose generic Jacobian rank is controlled at 0, 1 or 2, then invokes specialization theorems (Hilbert irreducibility or thin-set avoidance) to produce infinitely many distinct K-curves. No equation or definition equates the specialized rank to a fitted parameter or to the generic rank by construction; the upper bound on rank after specialization is supplied by an external argument (global descent or height control) rather than by re-using the input data. No self-citation is load-bearing for the uniqueness or existence claim, and no ansatz or renaming reduces the result to its own inputs. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the argument relies on standard facts about Jacobians of hyperelliptic curves and the Mordell-Weil theorem for abelian varieties over number fields; no new free parameters, ad-hoc axioms, or invented entities are indicated.

axioms (1)
  • standard math The Jacobian of a hyperelliptic curve over a number field is an abelian variety whose Mordell-Weil rank is well-defined.
    Invoked implicitly when stating that the Jacobian has rank 0, 1, or 2.

pith-pipeline@v0.9.0 · 5654 in / 1377 out tokens · 86138 ms · 2026-05-19T13:01:32.205402+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust

    arXiv:2210.10730. [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language.J. Symbolic Comput., 24(3-4):235–265,

    work page arXiv

  2. [2]

    [BD11] Nils Bruin and Kevin Doerksen

    Computational algebra and number theory (London, 1993). [BD11] Nils Bruin and Kevin Doerksen. The arithmetic of genus two curves with (4,4)-split Jacobians.Canadian Journal of Mathematics, 63(5):992–1024,

    work page 1993

  3. [3]

    Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups

    [BM] Alex Bartel and Adam Morgan. Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups. arXiv:2508.14026. [BS15] Manjul Bhargava and Arul Shankar. Binary quadratic forms having bounded invariants, and the bound- edness of the average rank of elliptic curves.Annals of Mathematics, 181:191–242,

    work page arXiv

  4. [4]

    Kolyvagin’s work on modular elliptic curves in L-functions and arithmetic (Durham, 1989).London Mathematical Society Lecture Note Series, 153:235–256,

    [Gro91] Benedict Gross. Kolyvagin’s work on modular elliptic curves in L-functions and arithmetic (Durham, 1989).London Mathematical Society Lecture Note Series, 153:235–256,

    work page 1989

  5. [5]

    [KMR14] Zev Klagsbrun, Barry Mazur, and Karl Rubin

    arXiv:2509.24937. [KMR14] Zev Klagsbrun, Barry Mazur, and Karl Rubin. A Markov model for Selmer ranks in families of twists. Compositio Mathematica, 150:1077–1106,

    work page arXiv

  6. [6]

    [KP25] Peter Koymans and Carlo Pagano

    arXiv:2412.01768. [KP25] Peter Koymans and Carlo Pagano. Elliptic curves of rank one over number fields,

    work page arXiv

  7. [7]

    [KR89] Ernst Kani and Michael Rosen

    arXiv:2505.16910. [KR89] Ernst Kani and Michael Rosen. Idempotent relations and factors of jacobians.Mathematische Annalen, 284:307–327,

    work page arXiv

  8. [8]

    [Mil86a] James Milne

    arXiv:2405.09311. [Mil86a] James Milne. Abelian varieties. In Joseph H. Silverman Gary Cornell, editor,Arithmetic Geometry, New York, USA,

    work page arXiv

  9. [9]

    Hasse principle for intersections of two quadrics via Kummer surfaces

    [MS] Adam Morgan and Alexei Skorobogatov. Hasse principle for intersections of two quadrics via Kummer surfaces. arXiv:2407.16459. [Mum74] David Mumford. Prym varieties i. InContributions to Analysis, pages 325–350. Academic Press,

    work page arXiv

  10. [10]

    [Sch95] Edward F

    arXiv:2506.17605. [Sch95] Edward F. Schaefer. 2-descent on the Jacobians of hyperelliptic curves.Journal of Number Theory, 51(2):219 – 232,

    work page arXiv

  11. [11]

    [Smi22b] Alexander Smith

    arXiv:2207.05674. [Smi22b] Alexander Smith. The distribution ofℓ∞-Selmer groups in degreeℓtwist families II,

    work page arXiv

  12. [12]

    [Smi25] Alexander Smith

    arXiv:2207.05143. [Smi25] Alexander Smith. The Birch and Swinnerton-dyer conjecture implies Goldfeld’s conjecture,

    work page arXiv

  13. [13]

    [SS03] Edward Schaefer and Michael Stoll

    arXiv:2503.17619. [SS03] Edward Schaefer and Michael Stoll. How to do ap-descent on an elliptic curve.Transactions of the American Mathematical Society, 356(3):1209–1231,

    work page arXiv

  14. [14]

    [Zyw25c] David Zywina

    arXiv:2505.16960. [Zyw25c] David Zywina. There are infinitely many elliptic curves over the rationals of rank 2,

    work page arXiv

  15. [15]

    Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Email address:stevangajovic@gmail.com, swpark2008@gmail.com

    arXiv:2502.01957. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Email address:stevangajovic@gmail.com, swpark2008@gmail.com

    work page arXiv