Tamagawa ratios and unbounded Selmer moments

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.

• The paper presents a new framework using the Greenberg–Wiles formula to link Tamagawa ratios with Selmer moment behavior in families of abelian varieties.
• It employs refined sieve methods and model theory to derive explicit asymptotic bounds and logarithmic growth laws for Selmer group moments.
• The results unify previous findings in arithmetic statistics, providing practical criteria for predicting when Selmer sizes remain bounded or diverge.

Tamagawa Ratios and Unbounded Selmer Moments: A Formal Summary

Introduction and Context

The paper "Tamagawa ratios and unbounded Selmer moments" (2606.31649) introduces a conceptual and quantitative framework for predicting when families of Selmer groups—arising in the context of abelian varieties over global fields—exhibit bounded versus unbounded average size. The approach is grounded in analyzing Tamagawa ratios via a refined application of the Greenberg–Wiles formula, linking global Selmer group sizes to products of local invariants.

A crucial motivation stems from arithmetic statistics and the study of the distribution of Selmer groups in families of elliptic curves, with reference to conjectures of Poonen–Rains, and subsequent explicit computations in several settings by Bhargava, Shankar, and collaborators. The authors generalize previous findings on the average size and moments of Selmer groups, especially those that displayed behavior sharply deviating from expected boundedness, such as unbounded growth in certain quadratic twist families.

Heuristic Framework and Main Conjecture

At the heart of the methodology is the Greenberg–Wiles formula, which, for a finite $G_F$-module $M$ with appropriate local conditions, yields the global-to-dual Selmer group size ratio as a product of Tamagawa local factors:

$$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$

where $T_v(M)$ encodes the dimension of local conditions modulo invariants.

The authors present Heuristic 1: in natural families of such Galois modules (with constant cardinality), the moments of the ratio $\#\mathrm{Sel}\,M/T(M)$ should be uniformly bounded, and the growth of Selmer moments is essentially controlled by that of the Tamagawa products. Thus, unbounded average size or moments of Selmer groups are predicted precisely in those settings where the Tamagawa ratios themselves have unbounded averages.

For geometric families of elliptic curves (parameterized by polynomials in several variables with integer coefficients), Conjecture 1 asserts a precise logarithmic growth law for the $\kappa$-th moment:

$$\frac{1}{\# A_{\leq H}}\sum_{E \in A_{\leq H}} \left(\# \mathrm{Sel}_\ell\, E\right)^{\kappa} \asymp (\log H)^{\beta(\kappa)},$$

where $\beta(\kappa)$ is computed via maximization over rational subgroups and associated local data (see the main text for explicit algorithms).

Technical Innovations

Greenberg–Wiles Formula as Predictive Tool

The key insight is that $T(M)$, as a product of local Tamagawa factors, produces a purely local lower bound for Selmer group sizes. In most "natural" settings, this lower bound is conjectured to tightly control the global Selmer size, up to moments of bounded order. The characterization extends Cassels' original philosophy—regarding isogeny Selmer groups and Tamagawa numbers—to broader and more technical contexts.

Sieve Methods and Gridding

The proof architecture refines previous methodologies by synthesizing sieve approaches (including techniques from Erdős, Shiu, and Nair–Tenenbaum) with a new "gridding" formalism for partitioning parameter space. The resulting framework allows for effective estimates on the distribution of the product of local factors (primes of bad reduction, etc.) necessary for bounding Tamagawa ratios across families.

Constant Galois Modules and Effective Equidistribution

A significant technical advance is the theorem covering families where the underlying Galois module is constant, but the local conditions vary. Under an effective equidistribution hypothesis (involving explicit error terms for local data across primes), sharp upper bounds for Selmer moments in terms of the Tamagawa moments are obtained. This effective equidistribution, vital for the approach, is established via algebraic and model-theoretic methods, including recourse to Denef–Pas quantifier elimination.

Principal Results

Main Theorem (Constant Module Case)

If the family of decorated $G_Q$-modules $M$0 is constant (i.e., $M$1 for all $M$2) and the local conditions are effectively equidistributed, then for all moments $M$3,

$M$4

for some $M$5, and

$M$6

This asserts that the Tamagawa moments control global Selmer group moments up to doubly exponential loss in $M$7, corresponding to the expected (and in some cases, provably sharp) growth.

Explicit Examples and Applications

The framework is applied to recover and generalize a number of key recent results:

• In classical quadratic twist families of elliptic curves, boundedness or unboundedness of Selmer moments is shown to be controlled by Tamagawa ratios, with precise logarithmic asymptotics.
• For families of curves with nontrivial isogeny structure at small primes (e.g., curves with full 2-torsion or with special isogenies), the moments of Selmer group size are computed explicitly, generalizing and systematizing earlier ad hoc approaches.
• The method further handles higher-dimensional families involving abelian varieties over number fields, and gives explicit bounds for moments of $M$8-torsion in class groups of quadratic fields, via an analogous local-to-global structure.

Notable Explicit Results

• For curves of the form $M$9, the $$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$0-Selmer rank moments grow as $$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$1 with $$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$2 the height parameter, reproducing Yu's earlier asymptotic and identifying the precise mechanism (Tamagawa ratios) for such growth.
• In certain families with full $$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$3-torsion structure, the expected size of the $$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$4-Selmer group diverges ($$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$5), but the probability of large $$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$6-Selmer rank decreases superexponentially, demonstrating subtle behavior governed by local data.

Theoretical and Practical Implications

These results have profound implications:

Theoretical:

• They strongly support the view that in generic or "random" families, the distribution of Selmer group sizes, moments, and related arithmetic objects is dictated, up to extremely small error, by products of local invariants—encoded by Tamagawa ratios.
• The framework unifies previously disparate phenomena (bounded vs. unbounded moments; behavior across isogeny classes; twist vs. generic families) under a single paradigm rooted in local-global duality.
• The use of model theory (Denef–Pas language) to transfer geometrical questions about local conditions to countable, computable, and effective problems is novel and likely to impact computational aspects of arithmetic statistics.

Practical/Future Directions:

• The method provides a practical algorithm for predicting, for new families (even those not yet studied), whether moments of Selmer group size will be bounded or diverge, and gives explicit rates of divergence where relevant.
• It is likely to guide the construction of counterexamples or confirmatory families for refined conjectures in arithmetic statistics, e.g., higher-$$\frac{\# \mathrm{Sel}\, M}{\# \mathrm{Sel}\, M^\vee} = T(M) = \frac{\# H^0(G_F,M)}{\# H^0(G_F, M^\vee)} \prod_v T_v(M),$$7 Selmer distributions or deviations from random matrix heuristics.
• Future work may extend these results to families with non-constant Galois modules or those parameterized by more arbitrary base schemes, as well as probing fine-scale distributional properties beyond moments.

Conclusion

This work rigorously demonstrates that Tamagawa ratios, accessible via the Greenberg–Wiles formula, sharply explain and predict the phenomenon of unbounded Selmer moments across natural families of abelian varieties and related Galois modules. Through an overview of sieve methods, local-global duality, and model theory, the authors resolve long-standing questions concerning the distribution of Selmer groups and provide a broadly applicable methodology for future research in arithmetic statistics.