An algorithm to compute Selmer groups via resolutions by permutations modules

CANARI; Fabrice Etienne (UB; IMB)

REVIEW 3 major objections 6 minor 18 references

An algorithm to compute Selmer groups via resolutions by permutations modules

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims an algorithm that computes the Selmer group of any finite Galois module from S-unit groups, using resolutions whose morphisms are Hecke operators.

desk verdict Novel idea for computing Selmer groups via Hecke-operator resolutions, but the key local proposition has a genuine gap and Theorem 3.8 is not established as written. read the letter →

arxiv 2504.13506 v1 pith:65VNC3NH submitted 2025-04-18 cs.SC math.GRmath.NTmath.RT

Fabrice Etienne (UB , CANARI , IMB) This is my paper · ORCID

classification cs.SCmath.GRmath.NTmath.RT MSC 11R3411R3211Y40

keywords SelmergroupsGaloiscohomologyHeckeoperatorspermutationmodulesS-unitsfinitecomputationalnumbertheoryclass

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 claims that Selmer groups, the Galois-cohomology objects behind descent and ranks of elliptic curves, can be computed by a uniform algorithm for any finite Galois module over a number field. The route is to resolve the module by permutation modules whose differentials are Hecke operators, dualize, and then read off cohomology from S-unit groups of the fields fixed by the stabilizers appearing in the resolution. The load-bearing comparison is a new group $H^1_S(\mathcal{G}, M)$, built from S-units, which the author proves is exactly the Selmer group with unramified conditions outside $S$ once $S$ contains all primes dividing $|M|$ and spans the class group of the field $L_0$. If the proof is right, every step except computing fixed fields and S-units/class groups is polynomial, so Selmer computation is reduced to arithmetic in S-unit groups.

What carries the argument

The central object is a partial resolution of the finite Galois module $M$ by duals of permutation modules, with differentials expressed as sums of Hecke operators. Hecke operators are the morphisms attached by a natural isomorphism to double cosets $H\backslash G/J$ between fixed-point modules, and the paper uses two imported properties: every surjective map between permutation modules has a quasi-inverse $\Psi$ with $\Phi \circ \Psi = k \cdot \mathrm{id}$ for a positive integer $k$ dividing $|G|^2$, and Hecke maps send S-units to S-units. These properties make $H^1_S$ well-defined and allow equations to be lifted from value groups. A Tor-functor argument on $N$-torsion-free permutation modules identifies the ramified local cohomology, completing the proof that $H^1_S$ is the unramified Selmer group outside $S$.

What would settle it

Take a finite Galois module whose algorithm-4.2 resolution has a nontrivial kernel for $d_0$, choose a place $v$ not dividing $|M|$, and seek $x$ in the local S-unit group with the same valuation as $d_0(z)$ but not equal to $d_0(z)$; finding such an $x$ would contradict the proof of Proposition 3.7 and therefore the identification of $H^1_S$ with the unramified Selmer group in Theorem 3.8.

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Extended reading notes

Core claim

Theorem A states that there is an algorithm which, given a finite Galois module $M$, the finite image $G$ of the Galois action, and a Selmer system $\mathcal{L}$, outputs the Selmer group $\mathrm{Sel}_{\mathcal{L}}$. The algorithm first constructs a partial resolution $P_2 \to P_1 \to P_0 \to M^* \to 0$ by permutation modules with morphisms given by Hecke operators. Dualizing gives an exact sequence $0 \to M \to I_0 \to I_1 \to I_2$, and the paper defines $H^1_S(\mathcal{G}, M)$ as the kernel of the induced map on S-unit groups modulo the image of the previous S-unit map. Theorem 3.8 proves that $H^1_S$ is the Selmer group attached to the Selmer structure with unramified conditions outside $S$, provided $S$ contains all primes dividing $|M|$ and the primes above $S$ span the class group $\mathrm{Cl}(L_0)$. Since every Selmer group with prescribed local conditions lies inside such an $H^1_S$, the algorithm recovers $\mathrm{Sel}_{\mathcal{L}}$ as a kernel inside $H^1_S$; the complexity statement is polynomial except for oracles that compute fixed fields, S-units, and class groups.

Load-bearing premise

The construction rests on two statements imported without proof from the author's earlier preprint—that surjective Hecke-operator maps admit splitting quasi-inverses with constant $k$ dividing $|G|^2$, and that Hecke operators preserve S-units—and, in Proposition 3.7, on treating equal valuations as equality, which needs the finite kernel of $d_0$ to vanish.

Editorial extensions

If this is right

Selmer computation for any finite Galois module reduces, modulo fixed-field and S-unit/class-group oracles, to linear algebra in S-unit groups.
The group $H^1_S(\mathcal{G}, M)$ itself is a Selmer group, so unramified-outside-$S$ Selmer groups admit an explicit S-unit description.
The method covers arbitrary finite Galois modules, not only modules arising from elliptic curves.
As a corollary noted in the paper, every Selmer group lies in some finitely generated $H^1_S$, giving another proof that Selmer groups are finitely generated.
With oracles for fixed fields, S-units, and class groups, the whole algorithm runs in time polynomial in the size of the input and in $|M|$.

Reading between the lines

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

Not in the paper itself: if the Hecke-operator properties from reference [6] are supplied with complete proofs, the method likely gives a practical route to Selmer groups for Galois modules arising in modularity and deformation problems.
The author's closing remark suggests the same resolution machinery could compute Selmer-type subgroups of $H^2(\mathcal{G}, M)$; a natural test is to formalize the $H^2$ analogue.
Because $H^1_S$ is defined entirely from S-units, the equality in Theorem 3.8 suggests a way to compare different Selmer structures by changing $S$, which may simplify the subgroup search in algorithm 4.3.
The fixed-field oracle is the true bottleneck; replacing it with known polynomial-time algorithms for special Galois groups would make the complexity statement unconditional in those cases.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's my take on arXiv:2504.13506.

The paper proposes something genuinely new: a general algorithm to compute Selmer groups for arbitrary finite Galois modules and arbitrary Selmer systems, using resolutions whose morphisms are Hecke operators. The earlier algorithms (Bruin, Maistret-Shukla) were aimed at elliptic curves. The construction of H^1_S and the claim that it equals a Selmer group under mild hypotheses is a nice insight, and Algorithm 4.3 is clearly laid out. The author also states honestly that the implementation is future work and that two key propositions come from his own preprint [6].

The problems are in the proof of Proposition 3.7, which is load-bearing for Theorem 3.8. The stress-test note is correct: the argument that 'Im(d0)=Ker(d1) for the N-th roots of unity' comes from dualizing a short exact sequence modulo N, but the maps in the dual sequence do not line up with d0 and d1. The image of P1/N in P0/N is not the same as P'_0/N after reduction. For the trivial module Z/N, the resolution Z --N--> Z -> Z/N ->0 dualizes to 0->Z/N->K^× -^N-> K^× ->0, and on N-torsion Im(d0)=1 while Ker(d1)=μ_N. So the claimed equality is false. That invalidates the first inclusion H^1_units ⊆ Ker(Res), and with it Theorem 3.8. The second inclusion also has an overstatement ('again by injectivity, d0(z)=x' from equality of valuations), though that one is repairable if you work in the unit quotient.

There are smaller things. Lemma 2.2's proof contains a typo (f∘i=1 instead of 0) and the kernel-inclusion is backwards; the lemma itself is true for Hom(-, Kbar^×), so this is presentational. Proposition 3.2 has a step where k·x being an S-unit is used to conclude x is an S-unit; that only works if the prime divisors of k are in S, which is not stated. And the paper depends on [6] for two nontrivial properties of Hecke operators; that's a legitimate dependency, not circularity, but it makes the proof conditional on an unpublished preprint.

Bottom line: the central theorem is not established as written. The idea is good enough that I would send the paper to a careful referee, not desk reject. But the author needs to fix Prop 3.7 before this is citable. The right audience is people working on algorithmic Galois cohomology and arithmetic statistics; they will find the setup worth understanding, but they should not rely on Theorem A yet. I would not cite it in my own work in the next year.

Referee Report

3 major / 6 minor

Summary. The paper claims an algorithm (Theorem A) that, given a finite Galois module M over a number field K and a Selmer system L, computes the Selmer group Sel_L. The construction first builds a partial resolution of M by duals of permutation modules whose differentials are Hecke operators, then defines a group H^1_S from S-units of étale algebras. Theorem 3.8 asserts that if S contains the primes dividing |M| and spans Cl(L0), then H^1_S is exactly the Selmer group with no local conditions above S and unramified conditions outside S. Section 4 packages this into Algorithms 4.2 and 4.3 and states a conditional polynomial-time bound using oracles for S-units/class groups and fixed fields.

Significance. The proposed method is genuinely more general than existing Selmer-group algorithms for elliptic curves and, if correct, would be a valuable tool. The paper is clearly organized and the algorithmic idea is attractive; the complexity statement is explicit about which steps are not known to be polynomial. However, the proof of the central local statement (Proposition 3.7) contains a false exactness assertion, so Theorem 3.8 and hence Theorem A are not established in the present version.

major comments (3)

[Proposition 3.7, first inclusion] The proof reduces to the assertion that "for the N-th roots of unity, Im(d0) = Ker(d1)", justified by dualizing 0 -> P'_2/N -> P1/N -> P'_0/N -> 0. This does not follow: d0 on N-torsion is the map Hom(P0/N, mu_N) -> Hom(P1/N, mu_N), whereas exactness of the displayed sequence only controls Hom(P'_0/N) -> Hom(P1/N) -> Hom(P'_2/N). The assertion is false in general: for M = Z/N with trivial G-action and the resolution P2 = 0, P1 = Z --N--> P0 = Z, the induced maps on N-torsion are d0 = 0 and d1 = 0, so Im(d0|I0[N]) = {1} while Ker(d1|I1[N]) = mu_N. Since this exactness is used to discard the root of unity zeta_N and conclude that a local unit lies in B1_ram, the inclusion H^1_units,v subset of Ker(Res) is not proved as written. This is load-bearing because Proposition 3.7 is the key input to Theorem 3.8.
[Proposition 3.7, second inclusion] After deriving val(Res(d0(z)x^{-1})) = 0, the proof concludes "so, again by injectivity, d0(z) = x". Injectivity of d0 on the valuation lattice gives only equality of valuations, not equality of the elements in the multiplicative groups. The desired conclusion can likely be obtained by working modulo units, since val(Res(d0(z)x^{-1})) = 0 already says that x*d0(z)^{-1} is a unit, but the argument as written does not say this.
[Definition 3.1 and Proposition 3.2] Definition 3.1 and the proof of Proposition 3.2 rely on Propositions 1.3 and 1.4, which are imported from the author's preprint [6] without proof. Proposition 1.4 is essential for H^1_S to be well-defined, since Hecke operators must send S-units to S-units, and Proposition 1.3 is used in the proof of the injection H^1_S into H^1. The manuscript should either prove these facts or state explicitly and verify the hypotheses under which [6] applies; as it stands, a central part of the construction depends on an unverified source.

minor comments (6)

[Lemma 2.2] The proof contains two evident typos: "i composed with s = 0" should be "s composed with i = 0", and "f composed with i = 1" should be "f composed with i = 0". The claim that i^* is surjective because i is injective also deserves a justification using divisibility of K^times; the lemma is true, but the proof as written is incomplete.
[Definition 3.3] The displayed definitions of Z^1_ram and B1_ram appear to interchange the indices of d0 and d1: Z^1_ram should be a kernel inside (L1 tensor K_v^ur)^times and B1_ram an image from (L0 tensor K_v^ur)^times, consistent with the usage in the proof of Proposition 3.7.
[Proposition 3.2] The proof has a variable-name inconsistency: "Since x is in B1" should refer to y. The saturation argument for Z^times_{S,L0} should also be spelled out explicitly.
[Definition 3.1 / Proposition 3.6] Definition 3.1 calls S a set of prime numbers, but S-units and the condition that S spans Cl(L0) require S to be a set of places or prime ideals; this should be clarified.
[Proposition 3.6] The proof that local adjustments can be patched globally is only a sketch: the reduction to fractional ideals and the use of the fact that S spans Cl(L0) should be written out as a weak-approximation or Chinese-remainder argument.
[Algorithm 4.3] The phrase "the smallest set S" may not be well-defined, since several sets can satisfy the listed conditions; the algorithm should say "choose a finite set S satisfying the following conditions".

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the Selmer computation is derived from stated assumptions; the main self-citation dependency is a verification issue, not a circularity.

full rationale

The paper's central correctness statement, Theorem 3.8, is a genuine deduction rather than a restatement of an input: H^1_S is defined as a quotient built from S-unit groups of étale algebras and the maps d0, d1, while the Selmer group Sel_L is defined independently from local conditions in Galois cohomology. Proposition 3.6 proves H^1_S = {x in H^1 : Res_v(x) in H^1_units,v for all v not in S}, and Proposition 3.7 identifies H^1_units,v with the kernel of restriction to H^1_ram. Neither side of these equalities is defined in terms of the other, and no parameter is fitted to the target Selmer group. The only self-citation that is load-bearing is the import of Propositions 1.3 and 1.4 from the author's own preprint [6]: Proposition 1.4 is used to make H^1_S well-defined, and Proposition 1.3 is used in Proposition 3.2 to embed H^1_S into H^1. These are prior results with stated assumptions about Hecke operators and finite groups, not assumptions of the Selmer statement itself, so relying on them is normal mathematical dependence rather than circular reasoning. However, because they are not proved in this paper and come from the same author's unpublished preprint, the correctness of H^1_S's construction is a verification risk; the paper itself also flags the unresolved complexity of fixed-field computation in Remark 4.6. The apparent gap in Proposition 3.7, where the proof asserts Im(d0)=Ker(d1) for N-th roots of unity by dualizing an exact sequence, is a correctness issue, not a circular-reasoning issue. Overall, the derivation does not reduce by construction to its inputs, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The paper depends on the unverified prior work [6] and on standard cohomology facts. The number-theoretic content (class groups, S-units) is taken as input from oracles.

assumptions (5)

domain assumption Propositions 1.3 and 1.4 from [6]: existence of a splitting morphism Psi with Phi circ Psi = k times id, k dividing |G|^2, and preservation of S-units by Hecke operators.
These are imported verbatim from the author's own preprint [6] and are load-bearing: Proposition 1.4 makes H^1_S well-defined (Def 3.1), and Proposition 1.3 is used in the proof of Proposition 3.2.
standard math Hom_Z(-, K^times) is exact because K^times (multiplicative group of the algebraic closure) is divisible, hence injective as a Z-module.
Used in Lemma 2.2 to dualize the resolution.
standard math Hilbert 90: H^1(G_L, K^times) = 0 for finite extensions L/K.
Used in Proposition 2.4 to compute H^1(G, M).
standard math The G-invariants of Ind_{G_L}^G K^times equal L^times, and I_i = plus L_{i,j}^times (Proposition 2.3).
Cited to Voskresenskii [13]; used throughout the computation of d0^G and d1^G.
standard math S-units Z^times_{L,S} are saturated in L^times, so k times x being an S-unit implies x is an S-unit.
Used in the proof of Proposition 3.2.

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Cite this review

Pith. "Pith review of An algorithm to compute Selmer groups via resolutions by permutations modules." pith.science (2026). https://pith.science/paper/65VNC3NH

@misc{pith2026250413506,
  author       = {Pith},
  title        = {Pith review of: An algorithm to compute Selmer groups via resolutions by permutations modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/65VNC3NH}},
  note         = {Machine review of arXiv:2504.13506}
}

abstract

Given a number field with absolute Galois group $\mathcal{G}$, a finite Galois module $M$, and a Selmer system $\mathcal{L}$, this article gives a method to compute Sel$_\mathcal{L}$, the Selmer group of $M$ attached to $\mathcal{L}$. First we describe an algorithm to obtain a resolution of $M$ where the morphisms are given by Hecke operators. Then we construct another group $H^1_S(\mathcal{G}, M)$ and we prove, using the properties of Hecke operators, that $H^1_S(\mathcal{G}, M)$ is a Selmer group containing Sel$_\mathcal{L}$. Then, we discuss the time complexity of this method.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [6]

Computing class groups by induction with generalised norm relations

Fabrice Etienne. Computing class groups by induction with generalised norm relations. 2025. arXiv: 2411.13124v2 [math.NT] . url: https://arxiv.org/abs/2411.13124v2

work page Pith review arXiv 2025

[2] [1]

On average sizes of Selmer groups and ranks in families of elliptic curves having marke d points

Manjul Bhargava and Wei Ho. On average sizes of Selmer groups and ranks in families of elliptic curves having marke d points. Preprint, arXiv:2207.03309 [math.NT] (2022). 2022. url: https://arxiv.org/abs/2207.03309

arXiv 2022

[3] [2]

L-functions and Tama- gawa numbers of motives

Spencer Bloch and Kazuya Kato. L-functions and Tama- gawa numbers of motives . English. The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 333-400 (1990). 1990

work page 1990

[4] [3]

Extensions and torsors for ﬁnite group schemes

Peter Bruin. Extensions and torsors for ﬁnite group schemes

work page

[5] [4]

Homological algebra

Henri Cartan and Samuel Eilenberg. Homological algebra. English. Paperback ed. Princeton, NJ: Princeton University Press, 1999. isbn: 0-691-04991-2

work page 1999

[6] [5]

Advanced topics in computational number the- ory

Henri Cohen. Advanced topics in computational number the- ory. English. Vol. 193. Grad. Texts Math. New York, NY: Springer, 2000. isbn: 0-387-98727-4. doi: 10.1007/978-1-4419-8489-0

doi:10.1007/978-1-4419-8489-0

[7] [7]

Constructions using Galois Theory

Claus Fieker and Nicole Sutherland. Constructions using Galois Theory . 2022. arXiv: 2010.01281 [math.NT] . url: https://arxiv.org/abs/2010.01281

work page Pith review arXiv 2022

[8] [8]

Holt, Bettina Eick, and Eamonn A

Derek F. Holt, Bettina Eick, and Eamonn A. O’Brien. Hand- book of computational group theory. English. Discrete Math. Appl. (Boca Raton). Boca Raton, FL: Chapman & Hal- l/CRC Press, 2005. isbn: 1-58488-372-3

work page 2005