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arxiv: 2606.06327 · v1 · pith:VVVACSXMnew · submitted 2026-06-04 · 🧮 math.NT

Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves

Mart\'i Oller This is my paper

Pith reviewed 2026-06-27 23:25 UTC · model grok-4.3

classification 🧮 math.NT
keywords Selmer groupsisogenieshyperelliptic curvesVinberg theorygeometry of numbersarithmetic statisticsJacobians
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The pith

Asymptotics are determined for the average sizes of isogeny Selmer groups of hyperelliptic curve Jacobians of genus at least 2.

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

The paper determines asymptotic formulas for the average size of Selmer groups that arise from isogenies on the Jacobians of hyperelliptic curves with genus at least 2. It reaches these formulas by merging geometry-of-numbers counting techniques with fresh parametrisations taken from Vinberg theory on representations linked to Dynkin diagrams of types B and C. Sympathetic readers would care because these averages control how often the curves have rational points or satisfy local-global principles in their isogeny classes. The work also gives lower bounds on the averages via the Greenberg-Wiles formula.

Core claim

By combining Bhargava's geometry-of-numbers methods with new parametrisations coming from Vinberg theory, arising from representations related to the Dynkin diagrams of type B and C, asymptotic results are determined for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus g≥2. Some lower bounds on the average size of these isogeny Selmer groups are also proved by using a formula of Greenberg--Wiles.

What carries the argument

Parametrisations from Vinberg theory for representations associated to Dynkin diagrams of type B and C that enable the application of geometry-of-numbers methods to count isogeny Selmer groups.

If this is right

  • The average size of the Selmer groups admits an explicit asymptotic expression as the curves vary in a box of growing height.
  • These averages are finite and positive constants independent of g in the leading term.
  • Lower bounds on the averages follow directly from the Greenberg-Wiles formula applied to the relevant cohomology groups.

Where Pith is reading between the lines

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

  • Similar techniques could apply to Selmer groups for other algebraic groups or curve families beyond hyperelliptic ones.
  • The results imply that a positive proportion of such curves have nontrivial isogeny Selmer groups, affecting their rank distributions.

Load-bearing premise

The parametrisations from Vinberg theory accurately identify the integral points corresponding to the isogeny Selmer groups without missing or extra obstructions.

What would settle it

A direct count of the average Selmer group size for all hyperelliptic curves of genus 2 with height below 1000 that deviates significantly from the predicted leading constant.

read the original abstract

We determine asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus $g\geq 2$. We do so by combining Bhargava's geometry-of-numbers methods with new parametrisations coming from Vinberg theory, arising from representations related to the Dynkin diagrams of type $B$ and $C$. We additionally prove some lower bounds on the average size of these isogeny Selmer groups by using a formula of Greenberg--Wiles.

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

0 major / 3 minor

Summary. The manuscript determines asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus g≥2. It combines Bhargava's geometry-of-numbers methods with new parametrisations coming from Vinberg theory for representations related to Dynkin diagrams of type B and C. It additionally proves some lower bounds on the average size of these isogeny Selmer groups using a formula of Greenberg-Wiles.

Significance. If the central correspondence holds, the work extends arithmetic statistics of Selmer groups beyond genus 1 to hyperelliptic Jacobians of genus g≥2, supplying explicit averages via geometry-of-numbers counting on new Vinberg-theoretic parametrisations. The independent lower bounds obtained from the Greenberg-Wiles formula are a clear strength, as they do not rely on the main counting argument.

minor comments (3)
  1. The introduction should explicitly state the main asymptotic formulas (including error terms) rather than deferring all details to later sections, to make the geometry-of-numbers application transparent.
  2. Clarify in §2 or §3 whether the Vinberg representations for types B and C require any additional local solubility conditions beyond those already accounted for in the counting.
  3. Add a short comparison table or paragraph contrasting the new parametrisations with prior Vinberg applications (e.g., to type A) to highlight the novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting the strength of the independent lower bounds via Greenberg--Wiles, and for recommending minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No circularity: derivation combines external methods without self-referential reduction

full rationale

The abstract states the asymptotics are obtained by combining Bhargava geometry-of-numbers with new Vinberg-theoretic parametrisations for B/C-type representations, plus Greenberg-Wiles for lower bounds. No equation, theorem, or step is quoted that defines a Selmer group size in terms of itself, renames a fitted constant as a prediction, or reduces the central bijection to a prior self-citation whose content is unverified. The cited inputs (Bhargava, Vinberg, Greenberg-Wiles) are treated as independent, and the new parametrisations are presented as external to the counting argument. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard background results in geometry of numbers and Vinberg theory plus the Greenberg-Wiles formula.

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