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Selmer group associated to the Chow group of certain codimension two cycles

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Pith's one-line read The paper proves that for the self-product of a CM elliptic curve over a number field, the Galois-fixed n-torsion cokernel of flat pullback on algebraically trivial codimension-two cycles is a profinite group.

desk verdict A plausible extension of Mildenhall's theorem is undermined by a misapplication of finiteness to the algebraic closure, leaving the central claim unproven. read the letter →

arxiv 1908.06424 v3 pith:6JRT7YTX submitted 2019-08-18 math.NT math.AGmath.KT

Kalyan Banerjee , Kalyan Chakraborty This is my paper · ORCID
classification math.NTmath.AGmath.KT MSC 11G0511G1514C2514K22
keywords complexmultiplicationellipticcurveSelmergroupTate-ShafarevichChowAbelianvarietycodimensiontwocyclesprofinite
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

For a surface obtained as the self-product of an elliptic curve with complex multiplication over a number field, the paper proves a finiteness principle for codimension-two cycles. The object studied is the cokernel, at the level of n-torsion and Galois-fixed cycles, of the flat pullback from a smooth integral spread to the generic fiber: this cokernel is shown to be a profinite group, an inverse limit of finite groups. The proof constructs an n-Selmer group attached to the kernel of that pullback and shows all its elements are unramified outside finitely many places. If correct, the result says that the arithmetic of algebraically trivial 2-cycles in this family is controlled by cohomological data of the same flavor as the Selmer groups bounding abelian varieties.

What carries the argument

The central object is the Galois module $\Sigma_{\bar K}$, the kernel of $A^2(\mathscr{X}_{\bar K})\to A^2(X_{\bar K})$, together with its associated $n$-Selmer group $S_n(\Sigma_{\bar K})$, defined as the kernel of the global-to-local map $H^1(G,\Sigma_{\bar K}[n])\to \prod_v H^1(G_v,A^2(\mathscr{X}_{\bar K_v}))[n]$. The argument shows that every class in this Selmer group is unramified at all but finitely many places: at good primes $v\nmid n$, the class maps into inertia cohomology, and Raskind's specialization injectivity forces the image to vanish. Then, assuming $\Sigma_{\bar K}$ is profinite, Silverman's profinite unramified-cohomology lemma makes $S_n(\Sigma_{\bar K})$ profinite. Roitman's isomorphism $A^2(X)[n]\cong \operatorname{Alb}(X)[n]$ as Galois modules is the bridge that lets standard abelian-variety Selmer techniques be applied to the Chow group.

What would settle it

Take a concrete CM elliptic curve $E/K$, fix an integer $n$, and look for a class $\eta\in H^1(G,\Sigma_{\bar K}[n])$ whose image in every local group $H^1(G_v,A^2(\mathscr{X}_{\bar K_v}))[n]$ is zero but whose restriction to the inertia group $I_v$ is nonzero for infinitely many good primes $v\nmid n$. The theorem predicts no such class exists, so exhibiting one would disprove the profiniteness conclusion.

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Extended reading notes

Core claim

Let $E$ be an elliptic curve over a number field $K$ with complex multiplication by the ring of integers, discriminant $N$, and let $X=E\times E$. Choose a smooth spread $\mathscr{X}$ over $\mathcal{O}_K[1/6N]$, and let $A^2$ denote the group of algebraically trivial codimension-two cycles modulo rational equivalence. The paper's main theorem, Theorem 2.2, states that for the flat pullback $j^*:A^2(\mathscr{X})\to A^2(X)$, the quotient $\operatorname{im}(j^*)[n](K)/A^2(\mathscr{X})[n](K)$ is a profinite group. The proof introduces the $n$-Selmer group $S_n(\Sigma_{\bar K})$ attached to the kernel $\Sigma_{\bar K}$ of the restriction map after base change to the algebraic closure, proves that elements of this Selmer group are unramified outside a finite set of places, and then applies a profinite cohomology lemma. The conclusion is an arithmetic finiteness statement: the cokernel is not merely a torsion or divisible group but is built from finite layers in a compact inverse limit.

Load-bearing premise

The proof requires that the kernel of the restriction map after base change to the algebraic closure, the Galois module the paper calls $\Sigma_{\bar K}$, is profinite; the cited theorem invoked for this gives finiteness of the analogous kernel over the base field $K$, not over the algebraic closure.

Editorial extensions

If this is right

  • The same profiniteness conclusion holds for the Fermat quartic surface, since the analogue of Mildenhall's kernel-finiteness was proved there, as noted in Remark 2.4.
  • The quotient $\operatorname{im}(j^*)[n](K)/A^2(\mathscr{X})[n](K)$ embeds into $H^1(G,\Sigma_{\bar K}[n])$ and its image is contained in classes unramified outside a finite set of places, so the cokernel is governed by finitely many local conditions.
  • The construction associates to each $n$ an $n$-Selmer group for the restriction map on $A^2$, giving a Chow-group analogue of the Selmer group of an abelian variety.
  • Because the quotient is profinite, all of its arithmetic information is recoverable from its finite quotients, so the cokernel cannot contain an infinite divisible direct summand.
  • The theorem upgrades the known finiteness of the kernel of flat pullback to structural control of the cokernel on n-torsion Galois-fixed cycles.

Reading between the lines

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

  • Editorial inference: the proof scheme should extend to any smooth projective surface over a number field for which the restriction kernel on $A^2$ is profinite after base change and $A^2[n]$ is Galois-isomorphic to the n-torsion of an abelian variety; Remark 2.4 already exercises one such case.
  • Editorial inference: the theorem establishes profiniteness, not finiteness; computing $S_n(\Sigma_{\bar K})$ for small $n$ on a concrete CM elliptic curve could reveal whether the Chow-group Selmer group is finite like an abelian-variety Selmer group or has infinite profinite components.
  • Editorial inference: the Selmer-group construction is not obviously limited to $E\times E$; applying the same global-to-local map to higher Chow groups or motivic cohomology of codimension two would test whether the unramified-class mechanism is a general phenomenon for algebraic cycles.
  • Editorial inference: if a future strengthening replaced profiniteness of $\Sigma_{\bar K}$ by finiteness, the same proof would promote the conclusion from profinite to finite.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the flat pullback map j*: A^2(X) → A^2(X), where X is a smooth spread of the self-product E×E of a CM elliptic curve over the ring of integers localized away from 6N, and A^2 denotes the group of algebraically trivial codimension-two cycles modulo rational equivalence. It defines an n-Selmer group S_n(Σ_{\bar K}) as the kernel of H^1(G,Σ_{\bar K}[n]) → ∏_v H^1(G_v,A^2(X_{\bar K_v}))[n], and attempts to prove that S_n(Σ_{\bar K}) is profinite, and hence that im(j*)[n](K)/A^2(X)[n](K) is profinite. The proof follows the pattern of finiteness proofs for elliptic curve Selmer groups: it invokes Mildenhall's finiteness result for the kernel of restriction, Roitman's isomorphism between n-torsion of A^2 and Alb, and Raskind's injectivity of specialization to show that Selmer elements are unramified outside a finite set, then cites a profinite version of Silverman's Lemma X.4.3.

Significance. If the argument were correct, the theorem would be a nontrivial contribution: it would provide a Selmer-type profiniteness statement for the cokernel of flat pullback on algebraically trivial zero-cycles for self-products of CM elliptic curves, going beyond Mildenhall's finiteness of the kernel. The paper is concise and the analogy with elliptic curve Selmer groups is appealing. The construction of the Selmer group is a standard one and is not circular, and the main cited theorems (Mildenhall, Roitman, Raskind) are appropriate tools. However, as detailed below, the central proof rests on an unsupported profiniteness assertion, and there are additional gaps in the passage from the Selmer group to the desired quotient. The intended result is therefore not established by the manuscript.

major comments (3)
  1. [Section 2, exact sequence after Mildenhall's finiteness; final paragraph of proof of Theorem 2.2] The claim that Σ_{\bar K} is profinite by Mildenhall's theorem is unsupported. Mildenhall's cited theorem gives finiteness of the kernel Σ_K for the spread over O_K[1/6N], not of the base-changed kernel Σ_{\bar K} over the integral closure of O_K in \bar K. These are different groups, and Chow groups may acquire new kernel classes after scalar extension. Even if Mildenhall's finiteness held over every finite extension L/K, the group Σ_{\bar K} would be a filtered colimit of finite groups, which need not be profinite. Since Lemma 2.3 requires M to be a profinite G-module, its application to M=Σ_{\bar K} is unjustified.
  2. [Definition 2.1 and proof of Theorem 2.2 (use of Lemma 2.3)] The proof applies Lemma 2.3 to M=Σ_{\bar K}, but S_n(Σ_{\bar K}) is defined as a subgroup of H^1(G,Σ_{\bar K}[n]), not of H^1(G,Σ_{\bar K}). Even if Lemma 2.3 were applicable to Σ_{\bar K}, it would yield profiniteness of H^1(G,Σ_{\bar K};S), not of S_n(Σ_{\bar K}). To prove S_n profinite one would need the lemma for the module Σ_{\bar K}[n] (or a separate argument), and this is not what the manuscript does.
  3. [Proof of Theorem 2.2, final sentence] The step 'S_n(Σ_{\bar K}) is pro-finite, hence im(j*)[n](K)/A^2(X)[n](K) is pro-finite' is not justified. The quotient embeds into H^1(G,Σ_{\bar K}[n]) and, under the intended argument, into S_n, but an abstract subgroup of a profinite group need not itself be profinite unless it is closed (or otherwise known to be finite). No closedness or finiteness argument is supplied, so the final implication does not follow from the profiniteness of S_n alone.
minor comments (4)
  1. [Throughout, especially Definition 2.1 and final paragraph] The notation is inconsistent: the paper writes both S_n(Σ_{\bar K}) and S_n(Σ_K), and it defines S_n using H^1(G,Σ_{\bar K}[n]) while later invoking Lemma 2.3 with M=Σ_{\bar K}; these should be reconciled.
  2. [Section 2, paragraph after the exact sequence] The sentence 'Σ^G_{\bar K} is a subgroup of Σ_K. For our notational convenience we continue to denote Σ^G_{\bar K} as Σ_K' is confusing, because the immediately following cohomology group is H^1(G,Σ_{\bar K}), not H^1(G,Σ^G_{\bar K}); this notational shift should be clarified or removed.
  3. [Lemma 2.3] Lemma 2.3 is stated as a 'variance' of Silverman's Lemma X.4.3 without proof or a precise topology on H^1(G,M;S); the cited lemma concerns finiteness of Selmer groups for elliptic curves and does not directly justify the stated profiniteness assertion for arbitrary profinite G-modules.
  4. [Introduction and proof] There are several typographical errors, including 'flat-pullbcak' in the introduction, 'th ere' in the proof of Theorem 2.2, and 'variance of lemma' instead of 'variant of Lemma'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Selmer-group argument derives the profiniteness claim from external theorems, not from its own conclusion.

full rationale

The derivation chain in this paper is not circular. The target quotient im(j*)[n](K)/A2(X)[n](K) is embedded into H^1(G, Sigma_{\bar K}) via Galois cohomology of the exact sequence for the kernel of the flat pullback, and then S_n(Sigma_{\bar K}) is defined as the kernel of the map from H^1(G, Sigma_{\bar K}[n]) to the product of local H^1 groups. The paper then invokes a variant of Silverman's lemma (Lemma 2.3) to conclude that this Selmer group is pro-finite, using the claimed profiniteness of Sigma_{\bar K} from Mildenhall's theorem. None of these steps defines the conclusion into the inputs: S_n(Sigma_{\bar K}) is not defined to be pro-finite, and the quotient im(j*)[n](K)/A2(X)[n](K) is not used as a hypothesis for Lemma 2.3. The external results cited (Mildenhall, Roitman, Raskind, Silverman) are independent of the paper's conclusion, and there is no fitted parameter renamed as a prediction and no author-generated uniqueness theorem invoked to forbid alternatives. The skeptical objection that Mildenhall's theorem only gives finiteness of the kernel over the base field K, while Lemma 2.3 is applied to the base-changed kernel Sigma_{\bar K}, is a genuine support gap in the proof, but it is a non-circular gap: the unsupported assertion is an input assumption about profiniteness of a Galois module, not an equation or definition that reduces the theorem to itself. Accordingly, no circular step is present, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on several external results: Mildenhall's finiteness theorem, Roitman's theorem identifying torsion in Chow groups with torsion of the Albanese, Raskind's injectivity of specialization, and a stated lemma (2.3) on profinite cohomology. The proof does not establish the pro-finiteness of Σ_{\bar K}, which is needed to apply Lemma 2.3.

assumptions (4)
  • domain assumption Mildenhall's theorem: the kernel of the restriction map A2(X) → A2(X) over K is finite.
    Used at the start of Section 2 to define Σ_K and later to claim Σ_{\bar K} is pro-finite.
  • domain assumption Roitman's theorem: Alb(X)[n] and A2(X)[n] are isomorphic as Galois modules.
    Invoked to transfer finiteness from the Albanese to the Chow group in the proof of unramifiedness.
  • domain assumption Raskind's theorem: the specialization map on n-torsion of A2 is injective for good reduction places.
    Used to show that σ.z − z vanishes in the n-torsion of the local Chow group.
  • standard math Lemma 2.3 (variant of Silverman X.4.3): H^1(G, M; S) is pro-finite for M a pro-finite G-module and S finite.
    Stated without proof in the paper and applied to conclude S_n(Σ_{\bar K}) is pro-finite.

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Pith. "Pith review of Selmer group associated to the Chow group of certain codimension two cycles." pith.science (2026). https://pith.science/paper/6JRT7YTX

@misc{pith2026190806424,
  author       = {Pith},
  title        = {Pith review of: Selmer group associated to the Chow group of certain codimension two cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JRT7YTX}},
  note         = {Machine review of arXiv:1908.06424}
}
abstract

Let $X$ be a surface with geometric genus and irregularity zero which is defined over a number field $K$. Let $\mathscr{X}$ denote a smooth spread of $X$ over the spectrum of a Zariski open subset in the spectrum of the ring of integers and $A^2$ stands for the group of algebraically trivial cycles on schemes modulo rational equivalence. If $j^*: A^2(\mathscr{X})\to A^2(X)$ be the flat pull-back corresponding to the embedding $j:X\hookrightarrow \mathscr{X}$ then we prove that $\im(j^*)(K)/A^2(\mathscr{X})(K)$ is a torsion group. Here $\im(j^*)(K)$, $A^2(\mathscr{X})(K)$ stand for the cycles fixed under the action of the absolute Galois group.

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