The preprint gives conditional rules for carrying two properties of fine Selmer groups from a cyclotomic-like extension to a larger extension with several p-adic directions. The properties are finite generation of the relevant modules and pseudonullity, a technical condition in the paper's algebraic framework. In this setting, vertical means moving from the smaller extension to the larger one.
The main statements are conditional on a cyclotomic-like starting extension, Greenberg's finiteness conditions, and assumptions on the coefficient ring. The result on pseudonullity also requires the ring to be Cohen-Macaulay, a condition that appears in the theorem's hypotheses.
A map between two levels
At the center is a restriction map comparing fine Selmer groups over the two extensions. The proof uses Greenberg's method to bound its kernel and cokernel, the two parts of the comparison that the method tracks.
After passing to the Pontryagin dual, the map has a finitely generated torsion kernel and cokernel and is a pseudo-isomorphism. That means the two sides agree up to the controlled algebraic discrepancy allowed by a pseudo-isomorphism. This is the paper's vertical-control theorem.
A separate criterion handles a move from a quotient group to the full group. Under the stated Cohen-Macaulay coefficient-ring and regularity assumptions, pseudonullity of a quotient-level version called the H-coinvariants, taken over the algebra attached to the quotient group, implies pseudonullity of the original module over the full algebra.
What carries upward
The paper presents these steps as the basis for two propagation statements, B- and B+. B- says that if the dual fine Selmer group over the smaller extension is finitely generated over the stated coefficient ring, then the dual group over the larger extension is finitely generated over the algebra for its Galois group.
B+ concerns pseudonullity. When the coefficient ring is Cohen-Macaulay, pseudonullity over the smaller Iwasawa algebra, the algebra attached to that extension, implies pseudonullity over the larger one. Both implications are conditional, and the coefficient-ring requirement is part of the result.
The assumptions have a sharp edge
The preprint also sets a boundary. For the representation A = Qp/Zp over the cyclotomic extension, it states that Greenberg's finiteness condition fails and does not expect the vertical-control theorem to hold.
A stated counterexample makes the warning concrete: the Pontryagin dual of the restriction map has zero kernel but a cokernel that is not pseudonull. The comparison can therefore fail on the cokernel side even when the kernel is absent.
Applications across arithmetic settings
The framework is applied to the Tate motive written as A = μp∞ under the paper's splitting condition, (Spl). In that setting, the preprint states vertical control and both the B- finite-generation and B+ pseudonullity implications.
For elliptic curves with potentially good reduction at all p-adic places, it states vertical control and both implications. For general elliptic curves satisfying (Spl-) and (Large), it states vertical control and B- finite-generation propagation, and its general theorem also states B+ pseudonullity propagation.
The same set of conclusions is stated for classical cuspidal newforms of even weight and level prime to p: vertical control, B- finite-generation propagation and B+ pseudonullity propagation. For p-ordinary cuspidal Hida families that meet the stated irreducibility and even-weight conditions, the theorem gives vertical control and B- propagation; B+ propagation is stated when the coefficient ring is Cohen-Macaulay.
A conditional map, not a universal one
The document header identifies the work as an arXiv version 1 preprint in math.NT, dated 26 August 2026. No journal venue is listed in the supplied metadata.
Taken together, the theorems offer a conditional route for moving finiteness and pseudonullity upward. Their conclusions remain tied to the stated hypotheses, including Greenberg finiteness and, for B+, Cohen-Macaulayness. The supplied acknowledgement says the authors received help finding counterexamples and independently verified them; no funding source is reported.
Paper data and sources
Original title: Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions
Authors: Peikai Qi, Ruichen Xu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text