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arxiv: 2604.16978 · v1 · submitted 2026-04-18 · 🧮 math.NT

Geometry-of-numbers methods over global fields II: Coregular representations

Manjul Bhargava , Arul Shankar , Xiaoheng Wang This is my paper

Pith reviewed 2026-05-10 06:42 UTC · model grok-4.3

classification 🧮 math.NT
keywords geometry of numberscoregular representationsglobal fieldselliptic curvesSelmer groupsaverage rankshyperelliptic curvesJacobians
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The pith

Geometry-of-numbers methods extend to count orbits over any global field, bounding average ranks of elliptic curves and determining average Selmer sizes.

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

This paper shows how to adapt geometry-of-numbers techniques to count orbits with bounded invariants in coregular vector spaces when the base field is an arbitrary global field. The resulting orbit counts are applied to families of elliptic curves and Jacobians of hyperelliptic curves. A reader cares because this produces bounds on average ranks and exact averages for Selmer groups that hold over every number field or function field of characteristic not 2, 3, or 5. Previously such results were limited to the rationals, so this provides a uniform arithmetic picture across all global fields.

Core claim

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field F of characteristic not 2, 3 or 5.

What carries the argument

Coregular vector spaces whose orbits with bounded invariants can be counted using volume estimates from the geometry of numbers, now extended to global fields.

If this is right

  • Average ranks of elliptic curves over any such global field F are bounded.
  • Average sizes of the Selmer groups of these elliptic curves are determined explicitly.
  • The same bounds and averages hold for Jacobians of hyperelliptic curves over F.
  • The results apply uniformly to all global fields F of characteristic not 2, 3, or 5.

Where Pith is reading between the lines

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

  • The method suggests that similar orbit-counting arguments could apply to other arithmetic invariants arising from coregular actions.
  • Uniformity over global fields implies that rank distributions are comparable between number fields and function fields.
  • The characteristic restrictions might be removable with further analysis of bad primes.

Load-bearing premise

The coregular representations remain well-behaved and the geometry-of-numbers volume estimates extend without essential change when the base is switched from the rationals to an arbitrary global field F of characteristic not 2, 3 or 5.

What would settle it

Finding a global field F of suitable characteristic where an explicit count of orbits with small invariants deviates significantly from the predicted volume would show that the extension fails.

read the original abstract

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field $F$ of characteristic not $2$, $3$ or $5$.

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 / 2 minor

Summary. The manuscript develops geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field F. It applies these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field F of characteristic not 2, 3 or 5.

Significance. If the results hold, this extends geometry-of-numbers orbit counting from the rational case to arbitrary global fields via adelic quotients and local volume computations. The fact that local densities agree with the Q-case for all but finitely many places, leaving the Euler product for the main term unchanged, is a clear strength that preserves explicit main terms and error-term control. This yields uniform statements on average ranks and Selmer sizes that apply equally to number fields and function fields.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph explicitly contrasting the new adelic setup with the methods of the predecessor paper (part I) to highlight what changes and what remains unchanged.
  2. [§2] Notation for the height functions and the coregular invariants after base change to F could be collected in a single preliminary subsection for easier reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of extending geometry-of-numbers methods to arbitrary global fields, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; methods developed independently for global fields

full rationale

The paper develops new geometry-of-numbers techniques for counting orbits in coregular spaces over arbitrary global fields F (char ≠2,3,5) via adelic quotients and explicit local volume computations at places of F. These local densities are shown to match the rational case for all but finitely many places, allowing the Euler product for the main term to carry over directly. No equations reduce a claimed prediction or count to a fitted parameter from the same data, no uniqueness theorem is imported solely from prior self-citations as a load-bearing axiom, and no ansatz is smuggled in via citation. The central results on average ranks and Selmer sizes follow from the new counting estimates without definitional collapse or renaming of known patterns. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of coregular representations whose invariants control Selmer groups and on the validity of geometry-of-numbers volume estimates over arbitrary global fields; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Coregular representations and their invariant rings behave as in the rational case when base-changed to an arbitrary global field of good characteristic.
    The counting method and the application to Selmer groups presuppose that the algebraic-group-theoretic properties survive the base change.

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Reference graph

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