arxiv: 2605.09155 · v1 · submitted 2026-05-09 · 🧮 math.AG · math.LO

Recognition: no theorem link

A curve and its abstract generalized Jacobian

Assaf Hasson, Benjamin Castle, Ishai Dan-Cohen

Pith reviewed 2026-05-12 02:00 UTC · model grok-4.3

classification 🧮 math.AG math.LO

keywords generalized Jacobianabstract Jacobiancurve recoveryfunction fieldsGalois twistsBooher-Voloch conjecturealgebraic curvesZilber theorem

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The pith

The data of a smooth proper curve with a rational point and coprime effective divisor can be recovered from the abstract generalized Jacobian and its embedded curve points up to automorphism twist.

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

This paper proves that given a smooth proper curve C over a field k, together with a k-rational point c and an effective divisor m coprime to c, the group of algebraic closure points of the generalized Jacobian J_m together with the subset of those points coming from C minus the support of m determine the original triple (C, c, m) up to twisting by an automorphism of the algebraic closure. The argument generalizes Zilber's earlier recovery result for the ordinary Jacobian. A direct corollary, via prior work of Booher and Voloch, is that when k is finite the same data can be read off from L-functions attached to characters of certain Galois extensions of the function field of C. The result matters because it shows that the arithmetic structure of the generalized Jacobian encodes enough geometry to reconstruct the curve itself.

Core claim

To a smooth proper curve C over a field k equipped with a k-point c and an effective divisor m coprime to c, one may associate the abstract group J_m(k-bar) of k-bar-points of the generalized Jacobian, as well as a subset (C minus Supp(m))(k-bar) inside J_m(k-bar). We show that the data (C, c, m) can be retrieved from this subset up to a twist by an automorphism of k-bar, proving a conjecture of Booher and Voloch.

What carries the argument

The abstract generalized Jacobian J_m(k-bar) equipped with the natural subset of points coming from the curve minus the support of m.

If this is right

When the base field is finite, the data (C, c, m) can also be recovered from L-functions of characters of certain Galois extensions of the function field of C.
The recovery result extends the classical case of the ordinary Jacobian treated by Zilber.
Arithmetic invariants attached to the generalized Jacobian suffice to reconstruct the underlying geometric curve, point, and modulus.

Where Pith is reading between the lines

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

The result hints that anabelian-type information in the Jacobian can determine the curve up to isomorphism in a stronger sense than previously known.
Similar recovery statements might hold for other moduli problems or for curves over local fields.
The method could be tested computationally by checking whether small-genus curves over finite fields are distinguished by their generalized Jacobian data.

Load-bearing premise

The curve must be smooth and proper over k with a k-rational point and the divisor must be effective and coprime to the point, with the argument using the standard construction and properties of the generalized Jacobian.

What would settle it

Two distinct triples (C, c, m) and (C', c', m') whose generalized Jacobians over the algebraic closure are isomorphic as groups, with the curve-point subsets corresponding under the isomorphism, yet no automorphism of the algebraic closure maps one triple to the other.

read the original abstract

To a smooth proper curve $C$ over a field $k$ equipped with a $k$-point $c$ and an effective divisor $\mathfrak m$ coprime to $c$, one may associate the abstract group $J_{\mathfrak m}(\bar k)$ of $\overline k$-points of the generalized Jacobian, as well as a subset \[ \tag{*} \big(C\setminus \operatorname{Supp}(\mathfrak m)\big)(\bar k) \subset J_{\mathfrak m}(\bar k). \] We show that the data $(C,c,\mathfrak m)$ can be retrieved from (*) up to a twist by an automorphism of $\overline k$, proving a conjecture of Booher and Voloch. By a result of Booher and Voloch this shows that when $k$ is a finite field, the same data may also be retrieved from $L$-functions of characters of certain Galois extensions of the function field of $C$. The proof is a generalization of Zilber's well known work "A curve and its abstract Jacobian".

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

This extends Zilber's reconstruction from ordinary Jacobians to the generalized case and proves the Booher-Voloch conjecture with a direct argument.

read the letter

The main thing here is that they recover the curve C, the point c, and the modulus m from the abstract group of points on the generalized Jacobian together with the embedded subset from the curve minus the support of m. The recovery works up to automorphism of the algebraic closure, and this settles the Booher-Voloch conjecture. Over finite fields it also gives recovery from L-functions via the earlier Booher-Voloch result on characters of Galois extensions of the function field.

Referee Report

0 major / 3 minor

Summary. The paper proves a reconstruction theorem: given a smooth proper curve C over a field k with k-rational point c and effective divisor m coprime to c, the abstract group J_m(k-bar) of k-bar-points of the generalized Jacobian together with the embedded subset (C minus Supp(m))(k-bar) determines the triple (C, c, m) up to automorphism of k-bar. This generalizes Zilber's reconstruction theorem for ordinary Jacobians and proves a conjecture of Booher and Voloch; as a corollary, when k is finite the data can also be recovered from L-functions of characters of certain Galois extensions of the function field of C.

Significance. If the result holds, it provides a direct generalization of an important reconstruction theorem from algebraic geometry to the setting of generalized Jacobians, using the standard embedding via the divisor class map relative to c and m. The work credits the precedent of Zilber and the conjecture of Booher-Voloch, and the finite-field application via L-functions adds arithmetic interest. The hypotheses (smooth proper curve, k-point c, m coprime to c) are the minimal ones ensuring the generalized Jacobian is a smooth commutative group scheme with the stated generating subset.

minor comments (3)

§1 (Introduction): the statement of the main theorem should explicitly cross-reference the section containing the proof of the reconstruction (currently only alluded to as a generalization of Zilber).
Notation: the symbol frak m for the modulus is introduced in the abstract but its definition and coprimality condition with c should be restated at the beginning of §2 for readers who skip the abstract.
The proof sketch in the abstract mentions 'the group law and linear equivalence relations that characterize the image'; the manuscript should add a sentence in §3 clarifying which exact relations are used to distinguish the image from other subsets.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment, which accurately describe the content and context of our manuscript, including the generalization of Zilber's theorem and the application to L-functions. The recommendation of minor revision is noted. Since the report lists no specific major comments, we have no point-by-point items to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a reconstruction theorem by generalizing Zilber's known result on ordinary Jacobians to the generalized Jacobian setting, using the standard embedding of the curve into its generalized Jacobian via the divisor class map relative to the point c and modulus m. All steps rely on the classical properties of smooth proper curves, effective divisors coprime to c, and the group law on J_m, which are external to the paper and not defined in terms of the target reconstruction. No fitted parameters, self-definitional reductions, or load-bearing self-citations appear; the argument is a direct algebraic proof that remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard axioms and constructions of algebraic geometry for smooth proper curves and their generalized Jacobians; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard properties of smooth proper curves over a field and the existence and functoriality of their generalized Jacobians
The embedding of curve points into the generalized Jacobian and the recovery argument presuppose these foundational results.

pith-pipeline@v0.9.0 · 5482 in / 1288 out tokens · 67269 ms · 2026-05-12T02:00:55.492313+00:00 · methodology

discussion (0)

Reference graph

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