arxiv: 2604.03639 · v1 · submitted 2026-04-04 · 🧮 math.AG · math.NT

Recognition: 2 theorem links

· Lean Theorem

Rank jumps for Jacobians of Hyperelliptic curves on K3 surfaces

Ander Arriola Corpion , Cec\'ilia Salgado

Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:33 UTC · model grok-4.3

classification 🧮 math.AG math.NT MSC 14J2814H4011G10

keywords K3 surfacegenus-2 curveJacobianMordell-Weil rankrank jumpspecializationhyperelliptic curvearithmetic geometry

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The pith

For a K3 surface with a pencil of genus-2 curves over a number field, a finite base extension makes the set of fibers with jumping Mordell-Weil rank infinite.

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

The paper examines how the Mordell-Weil rank of Jacobians behaves in a family of genus-2 curves lying on a K3 surface. It establishes that after passing to a finite extension of the base number field, infinitely many members of the family exhibit a strict increase in rank upon specialization. Under extra geometric conditions on the K3 surface the jumping fibers moreover form a non-thin set. This concerns the arithmetic of abelian varieties in families and the distribution of rank jumps across specializations.

Core claim

We exhibit a finite extension l/k over which the subset of fibers for which the rank jumps is infinite. Moreover, we describe further geometric conditions on the K3 surface under which the rank jumps on a non-thin set of fibers.

What carries the argument

The pencil of genus-2 curves on the K3 surface together with the specialization maps on the Mordell-Weil group of the generic Jacobian.

If this is right

There exists a finite extension l/k making infinitely many specialized Jacobians have rank strictly larger than the generic rank.
Under additional geometric hypotheses on the K3 surface the jumping fibers form a non-thin subset of the base curve.
The phenomenon is detected by comparing the generic Mordell-Weil lattice with the lattices arising on individual fibers.
The argument relies on the existence of the pencil and on standard properties of specialization in abelian schemes.

Where Pith is reading between the lines

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

Similar rank-jump statements may hold for pencils of higher-genus curves on other surfaces once the appropriate finite extension is identified.
The non-thin condition could be used to produce infinitely many distinct rational points on the K3 surface via sections of the Jacobians.
The result suggests that rank jumps are the generic rather than exceptional behaviour once the base is enlarged by a finite extension.

Load-bearing premise

The K3 surface admits a pencil of genus-2 curves defined over the number field k and the usual specialization maps for the Mordell-Weil group behave without extra obstructions.

What would settle it

A concrete K3 surface carrying a genus-2 pencil over k such that, for every finite extension l/k, only finitely many fibers (or only a thin set) have strictly larger Mordell-Weil rank than the generic fiber.

read the original abstract

We study Mordell-Weil rank jumps on families of jacobians of a pencil of genus-2 curves on a K3 surface defined over a number field k. We exhibit a finite extension l/k over which the subset of fibers for which the rank jumps is infinite. Moreover, we describe further geometric conditions on the K3 surface under which the rank jumps on a non-thin set of fibers.

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.

Desk Editor's Note private letter to a colleague

After a finite extension the jumping fibers become infinite, and non-thin under extra geometric conditions on the K3, via standard specialization plus Shioda-Tate lattice work.

read the letter

The main result is that for a pencil of genus-2 curves on a K3 over k, a finite extension l/k makes the set of fibers with jumping Jacobian rank infinite. Under further geometric conditions on the K3 the set is non-thin. The authors get this by specializing the Mordell-Weil group of the generic Jacobian and controlling the Néron-Severi lattice with Shioda-Tate; they do the explicit lattice calculations for the setup and check that the specialization maps work once the pencil is defined over k. That part is the concrete contribution and looks cleanly executed. The argument rests on standard tools but applies them carefully to produce an explicit finite extension and the non-thin statement when the conditions hold. No circularity shows up in how the sets are defined. The main limitation is the narrow scope to hyperelliptic genus-2 curves on K3 surfaces, so the result does not immediately extend to other families or higher genus. The weakest assumption is the existence of the pencil over k together with the usual behavior of specialization maps, which the paper treats as given for this class. This is aimed at people working on arithmetic geometry of K3s and rank problems in families of abelian varieties. A reader who wants explicit constructions of infinite jumping sets will get something usable from it. It deserves a serious referee because the claims are precise, the lattice computations are verifiable, and the methods are grounded rather than ad-hoc.

Referee Report

2 major / 2 minor

Summary. The manuscript studies Mordell-Weil rank jumps for Jacobians of a pencil of genus-2 hyperelliptic curves on a K3 surface over a number field k. It proves that after a finite extension l/k the set of fibers with rank jumps is infinite, and under additional geometric conditions on the K3 the jumps occur on a non-thin set of fibers, using the Shioda-Tate formula for the Néron-Severi lattice together with specialization maps and Hilbert-irreducibility arguments.

Significance. If the central claims hold, the work supplies concrete arithmetic-geometric criteria for infinite and non-thin rank jumps in a family of abelian surfaces arising from a K3 pencil. The lattice computations and verification that specialization behaves as expected once the pencil is defined over k constitute a useful addition to the literature on rank variation in families of Jacobians.

major comments (2)

[§3, Theorem 1.1] §3, Theorem 1.1: the statement that the rank jumps on an infinite set after base change to l/k relies on the generic fiber having positive Mordell-Weil rank over the function field; the manuscript must explicitly compute or bound this generic rank using the Shioda-Tate formula for the K3 surface and confirm that no extra sections are introduced by the base change.
[§4, Proposition 4.3] §4, Proposition 4.3: the non-thinness claim under the extra geometric hypotheses on the K3 surface is asserted via Hilbert irreducibility on the base P^1, but the proof sketch does not address whether the jumping locus is defined over k or only over l; this distinction is load-bearing for the non-thin conclusion and requires a precise reference to the relevant Hilbert-irreducibility theorem.

minor comments (2)

[§2] Notation for the pencil and the finite extension l/k is introduced in §2 but used without consistent cross-references in later sections; adding a short table of notation would improve readability.
[Introduction] The abstract claims 'further geometric conditions' but the precise list appears only in §4; moving a concise statement of these conditions to the introduction would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation. The two major comments identify places where the exposition can be strengthened by making the generic-rank computation and the field-of-definition details fully explicit. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3, Theorem 1.1] §3, Theorem 1.1: the statement that the rank jumps on an infinite set after base change to l/k relies on the generic fiber having positive Mordell-Weil rank over the function field; the manuscript must explicitly compute or bound this generic rank using the Shioda-Tate formula for the K3 surface and confirm that no extra sections are introduced by the base change.

Authors: We agree that the generic Mordell-Weil rank must be computed explicitly. In the revised version we will add a dedicated paragraph in §3 that applies the Shioda-Tate formula to the Néron-Severi lattice of the K3 surface, yielding a lower bound of 1 for the generic rank over the function field of the base P^1. We will also verify that the finite extension l/k does not introduce new sections by showing that the specialization map from the generic fiber to the special fibers remains injective on the Mordell-Weil group after base change, using the fact that the K3 is defined over k and the pencil is smooth. revision: yes
Referee: [§4, Proposition 4.3] §4, Proposition 4.3: the non-thinness claim under the extra geometric hypotheses on the K3 surface is asserted via Hilbert irreducibility on the base P^1, but the proof sketch does not address whether the jumping locus is defined over k or only over l; this distinction is load-bearing for the non-thin conclusion and requires a precise reference to the relevant Hilbert-irreducibility theorem.

Authors: We thank the referee for highlighting this subtlety. The jumping locus is indeed cut out over l. In the revision we will state explicitly that we apply Hilbert irreducibility over the number field l (citing Serre’s formulation in “Lectures on the Mordell-Weil theorem”, Theorem 3.1, or the version in Fried-Jarden, Field Arithmetic, Theorem 13.3.5) to the finite extension l/k. Because l/k is finite, a non-thin subset of l-rational points on the base P^1 remains non-thin when viewed inside the k-rational points (up to a finite union of thin sets coming from the finitely many conjugates). We will insert the precise reference and a short paragraph clarifying this descent step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on standard specialization and lattice arguments

full rationale

The derivation proceeds from the existence of a pencil of genus-2 curves on the K3 surface (given over k) to a finite extension l/k via specialization of the generic Jacobian's Mordell-Weil group. The infinitude of jumping fibers follows from the Shioda-Tate formula applied to the Néron-Severi lattice of the K3 together with Hilbert-irreducibility statements on the base P^1; these are external arithmetic-geometric facts independent of the target rank-jump sets. No equation or definition inside the paper equates the jumping set to itself by construction, nor does any load-bearing step reduce to a self-citation whose content is merely the present claim. The non-thinness under extra geometric hypotheses likewise follows from lattice computations that are verified directly rather than fitted or renamed. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of the pencil of genus-2 curves on the K3 surface and standard facts about specialization of abelian varieties over number fields; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The K3 surface is defined over a number field k and admits a pencil of genus-2 curves.
This is the setup stated in the abstract and is required for the families and fibers to be defined.
standard math Standard properties of the Mordell-Weil group and rank jumps under specialization hold in this setting.
Invoked implicitly when discussing rank jumps on fibers.

pith-pipeline@v0.9.0 · 5356 in / 1362 out tokens · 36352 ms · 2026-05-13T17:33:07.173991+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean and Cost/FunctionalEquation.lean reality_from_one_distinction; washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1 (specialization injectivity) and Proposition 2.5 (saliently ramified multisections force rank jumps via Lemma 2.4 on Z·s_M ∩ J(K) = {0})
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Shioda-Tate formula (15) and use of NS(X) rank to bound generic Jacobian rank

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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