A mathematical preprint reports that several algebraic measures attached to Selmer groups can stay controlled when a chosen Z_p-extension is replaced by a sufficiently nearby one. Under the paper’s stated assumptions, the dual Greenberg and strict Selmer groups remain torsion, their μ-invariant does not increase, and their λ-invariant does not increase when the corresponding μ-invariants agree. The result is a conditional form of local stability: moving within the prescribed neighbourhood does not make these measures larger.
The symbols μ and λ are Iwasawa invariants, summary measures the paper uses to compare Selmer-group behaviour across extensions. The objects are p-ordinary two-dimensional p-adic Galois representations over a number field K, together with varying Z_p-extensions. A Z_p-extension is the paper’s setting for moving from finite layers to an infinite extension; a Greenberg neighbourhood contains extensions treated as sufficiently close to a chosen one.
Selmer groups are accompanied by fine Selmer groups, which impose the paper’s more specialised local conditions. The central question is whether torsionness and the Iwasawa-invariant bounds continue to hold as those mathematical extensions vary within a neighbourhood.
How the comparison works
To support the comparison, the authors relate each finite layer to the full extension with a restriction map. Under the stated hypotheses, the map has zero kernel at every finite layer. Its cokernel—the part left over on the target side—is finite and becomes uniformly bounded from some layer onward. This control theorem supplies the bridge between finite-level information and the nearby-extension statements.
The proof then places the relevant inverse systems—the linked, layer-by-layer algebraic objects—in Fukuda Λ-modules. On a sufficiently small neighbourhood, those modules have uniformly bounded parameters. A cited theorem applied to that structure yields cotorsionness and the μ and λ bounds. The argument therefore works through a shared algebraic framework for the neighbourhood, rather than treating each extension as an unrelated case.
For Greenberg or strict Selmer groups, the reference dual group must be Λ-torsion. When that and the theorem’s other hypotheses hold, the nearby dual groups remain torsion. The theorem’s direct μ conclusion is that the nearby value is no larger than the reference value. The λ conclusion has a tighter entry condition: the relevant μ-invariants must agree before the upper bound can be applied.
The same pattern in finer Selmer groups
The paper extends the comparison to S-fine Selmer groups. In that setting, it assumes local H^0-vanishing at ramified primes and Λ-torsionness for the reference dual S-fine group. For a finite Σ containing S, a W-neighbourhood keeps the enlarged dual group torsion and does not increase its μ-invariant. The corresponding λ-invariant is bounded above by the reference λ only when the μ-invariants match.
The fine-group proof has a more specific algebraic marker. Under the local vanishing condition, the dual S-fine Selmer groups in the W-neighbourhood form Fukuda Λ-modules with parameters (1, 1, 1). That structure underpins the torsion and μ result; it does not remove the matching-μ condition attached to the λ bound.
Greenberg fine Selmer groups follow an analogous pattern. When local H^0 vanishes at primes above p and the reference dual group is Λ-torsion, the finer neighbourhood preserves torsionness and gives a non-increasing μ bound. Its λ bound is likewise conditional under the theorem’s assumptions.
A narrower link to p-adic L-functions
The paper’s most delicate conclusion concerns the expected connection between an algebraic characteristic ideal and a conjectural p-adic L-function. For an elliptic curve with good ordinary reduction, if the Iwasawa Main Conjecture holds over the cyclotomic extension and the reference μ and λ invariants both vanish, the nearby characteristic ideals are generated by their corresponding p-adic L-functions.
This is a propagation result, not a general proof of the Main Conjecture. The cyclotomic case and the zero reference invariants are inputs to the conclusion; the theorem carries the relation into a neighbourhood around that case. The supplied result does not say that the same relation persists when those reference invariants do not vanish.
A worked example shows how the conditional statement is used. It takes p = 5, K = Q(i), and an elliptic curve over Q identified by the label [LMF26, 52.a2]. Over the relevant cyclotomic extension, the example reports zero algebraic and analytic μ and λ invariants. It also states that both sides of the Iwasawa Main Conjecture are units, so both generate the unit ideal.
The example is illustrative of the theorem’s framework rather than a replacement for its assumptions. Its zero-invariant outcome fits the special case in which the nearby characteristic-ideal conclusion is available, while the broader theorems still depend on their own hypotheses.
What the preprint does—and does not—claim
The conditions define the reach of the results. The general neighbourhood theorem is set for ordinary two-dimensional representations and a Λ-torsion reference group. The fine-Selmer theorems add local H^0-vanishing requirements—at ramified primes for the S-fine case and at primes above p for the Greenberg fine case.
Taken together, the authors present the work as an extension of local boundedness to ordinary two-dimensional representations and several fine Selmer groups. They interpret the characteristic-ideal result as evidence that the expected p-adic-L-function relation persists near the cyclotomic extension, subject to the stated assumptions.
Several questions remain open in the supplied analysis: whether the characteristic-ideal relation can be obtained without vanishing reference μ and λ; whether similar bounds extend beyond the ordinary two-dimensional representations and Selmer conditions studied here; and whether broader unconditional Main Conjecture settings can support the same nearby-extension conclusions.
The supplied metadata lists the document as arXiv:2608.20130v1, dated 20 Aug 2026. Its acknowledgments report HRI postdoctoral fellowship support for the first named author and NBHM postdoctoral fellowship support for the fourth named author.
Paper data and sources
Original title: Variation of Iwasawa Invariants for Ordinary Representations
Authors: Abhishek, Chandrakant Aribam, Shiva Barman, Sohan Ghosh
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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