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Tamagawa ratios and unbounded Selmer moments

T0 review · 2 major / 2 minor · reviewed 2026-07-01 · grok-4.3

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Pith's one-line read Local factors decide when Selmer group averages are unbounded

desk verdict Proves the constant-torsion case cleanly via Greenberg-Wiles local products, but the general characterization rests on an unproven sharpness assumption for the local lower bound. read the letter →

arxiv 2606.31649 v1 pith:IC3AA5EX submitted 2026-06-30 math.NT

Peter Koymans , Alexander Smith This is my paper
classification math.NT
keywords SelmergroupsabelianvarietiesTamagawaratiosGreenberg-WilesformulaaveragesizesGaloismodulesunboundedmoments
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a method based on local computations to determine if the average size of certain Selmer groups grows without bound across families of abelian varieties over the rationals. The method uses a formula that decomposes a ratio of group sizes into local pieces at each prime, giving a minimum possible size from those pieces alone. The authors propose that this minimum is usually achieved, allowing them to classify families by whether their local data forces the average to grow. They confirm this classification when the structure of the points of order l is the same for every variety in the family.

What carries the argument

Greenberg-Wiles formula expressing the Selmer-dual Selmer ratio as a product of local factors, which supplies a purely local lower bound for Selmer sizes.

What would settle it

For a geometric family where the local product bound remains bounded, compute the average l-Selmer size and check whether this average stays bounded.

Watch

Extended reading notes

Core claim

Using the Greenberg-Wiles formula, which writes the ratio of the order of an l-Selmer group to the order of its dual as a product over local factors, the authors obtain a purely local lower bound on the size of the Selmer group. They conjecture that this bound is asymptotically sharp for most members of geometric families of abelian varieties over Q. This leads to a conjectural criterion, in terms of the growth of these local factors, for when the average Selmer size is unbounded. The criterion is proven when the Galois representation on the l-torsion is constant in the family.

Load-bearing premise

The purely local lower bound for Selmer group size from the Greenberg-Wiles formula is close to the actual size most of the time.

Editorial extensions

If this is right

  • The average size of l-Selmer groups is unbounded precisely when the product of local factors grows unbounded, at least when the l-torsion Galois module is constant.
  • Tamagawa ratios enter the local factors and therefore control whether the average Selmer size is bounded or not.
  • The framework yields predictions for any geometric family once the local data at each prime is known.
  • When the conjecture holds, the moments of Selmer sizes are determined by the growth rate of the local product.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Numerical checks of average Selmer sizes in families with constant torsion could confirm the proven case for small l.
  • If the local bound controls the averages, then global phenomena like the class group contribute negligibly on average in these families.
  • The same local criterion might extend to predict boundedness of higher moments or the distribution of ranks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework using the Greenberg--Wiles formula to predict whether the average size of ℓ-Selmer groups in geometric families of abelian varieties over ℚ is bounded or unbounded. It gives a conjectural characterization of which such families have unbounded average ℓ-Selmer size for fixed ℓ, and proves the characterization holds when the ℓ-torsion Galois module is constant across the family. The key tool is the formula's expression of the Selmer/dual-Selmer ratio as a product of local factors, yielding a purely local lower bound that is conjectured to be close to sharp most of the time.

Significance. If the sharpness conjecture holds, the framework would supply a concrete, Tamagawa-ratio-based criterion for unbounded Selmer moments, advancing the study of average ranks in families of abelian varieties. The proven special case (constant torsion module) is a concrete advance that validates the approach in a non-trivial regime and demonstrates control via the product of local factors.

major comments (2)
  1. [§1] §1 (main conjecture statement): the conjectural characterization for families with varying ℓ-torsion is load-bearing on the assumption that the Greenberg--Wiles local lower bound is typically sharp, but the manuscript supplies no density statement, heuristic density, or independent verification of this sharpness outside the constant-module regime; this leaves the general claim resting on an unproven posit rather than a derived property.
  2. [Theorem (constant-module case)] Theorem (constant-module case): while the proof correctly shows that divergence of the product of local Tamagawa ratios implies unbounded average Selmer size when the torsion module is constant, the argument does not address how often the lower bound is achieved in the varying-torsion setting, which is required to extend the conclusion.
minor comments (2)
  1. The abstract could briefly indicate the concrete families to which the framework is applied, to clarify scope for readers.
  2. [Introduction] Notation for the local factors in the Greenberg--Wiles formula could be recalled explicitly in the introduction for accessibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The manuscript presents the general characterization as a conjecture relying on the typical sharpness of the local bound, and proves the result only in the constant-torsion case. We respond to the major comments below.

read point-by-point responses

  1. Referee: §1 (main conjecture statement): the conjectural characterization for families with varying ℓ-torsion is load-bearing on the assumption that the Greenberg--Wiles local lower bound is typically sharp, but the manuscript supplies no density statement, heuristic density, or independent verification of this sharpness outside the constant-module regime; this leaves the general claim resting on an unproven posit rather than a derived property.

    Authors: The conjecture for varying torsion is explicitly presented as such precisely because it depends on the typical sharpness of the Greenberg--Wiles lower bound, which is not proven in the manuscript. The paper states that this sharpness is conjectured to hold most of the time and supplies a proof only when the torsion module is constant (where the argument does not require the same density). No density statement is given because establishing one lies outside the scope of the work and would itself be conjectural. The presentation accurately reflects the logical status of the claim, so no revision is required. revision: no

  2. Referee: Theorem (constant-module case): while the proof correctly shows that divergence of the product of local Tamagawa ratios implies unbounded average Selmer size when the torsion module is constant, the argument does not address how often the lower bound is achieved in the varying-torsion setting, which is required to extend the conclusion.

    Authors: The theorem is stated and proved exclusively for the constant-module case; it makes no claim about the varying-torsion setting. The conjecture for families with varying torsion is formulated separately and is not asserted to follow from the theorem. Extending the conclusion to the varying case would indeed require additional control on the frequency with which the lower bound is achieved, which remains part of the conjecture. The proof for the constant-module case is complete as written. revision: no

standing simulated objections not resolved
  • Provision of a density statement, heuristic density, or independent verification of sharpness of the Greenberg--Wiles local lower bound in the varying ℓ-torsion regime (this is the reason the general statement is conjectural rather than proven).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; proven case uses external formula directly and general case is explicitly conjectural

full rationale

The paper applies the Greenberg-Wiles formula (an external result) to obtain a local lower bound on Selmer sizes and proves the characterization holds when the l-torsion module is constant across the family. The general characterization is stated as conjectural, resting on the explicit additional assumption that the local bound is typically close to sharp. No quoted step reduces a claimed prediction or result to an input by construction, self-definition, or load-bearing self-citation chain. The derivation for the proven regime is independent of the conjecture and does not rename or smuggle in prior results tautologically.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review based solely on abstract; full paper not available. The Greenberg-Wiles formula is treated as a standard tool. The main unproven input is the sharpness conjecture for the local lower bound.

assumptions (1)
  • standard math Greenberg-Wiles formula expresses Selmer/dual-Selmer ratio as product of local factors
    Invoked as the key tool in the abstract; treated as background result from prior literature.

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Pith. "Pith review of Tamagawa ratios and unbounded Selmer moments." pith.science (2026). https://pith.science/paper/IC3AA5EX

@misc{pith2026260631649,
  author       = {Pith},
  title        = {Pith review of: Tamagawa ratios and unbounded Selmer moments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IC3AA5EX}},
  note         = {Machine review of arXiv:2606.31649}
}
abstract

We develop a framework to predict whether a family of Selmer groups has average size that is bounded or unbounded. Applying this framework to certain geometric families of abelian varieties over $\mathbb{Q}$, we give a conjectural characterization of which such families have $\ell$-Selmer groups of unbounded average size for a given prime $\ell$. In the case that the $\ell$-torsion Galois module is constant across the family, we show that our characterization is correct. The key tool of our technique is the Greenberg--Wiles' formula, which expresses the ratio of the sizes of a Selmer group and the corresponding dual Selmer group as a product of local factors. This formula gives a purely local lower bound for the size of a Selmer group that we conjecture is close to sharp most of the time.

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