Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Selmer ranks in twists of CM abelian varieties

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For CM abelian varieties satisfying a simplicity hypothesis, Selmer ranks in p-th twists obey the symplectic or unitary distribution D^epsilon_q, giving unsolvability of most twisted Fermat curves along a fan structure.

desk verdict First symplectic/unitary Selmer-rank distribution for CM twists, but the load-bearing local quadratic-space structure is imported from the companion paper; a serious referee is needed. read the letter →

arxiv 2504.21274 v2 pith:M5UMQQ6B submitted 2025-04-30 math.NT

Jie Shu This is my paper · ORCID
classification math.NT
keywords abelianvarietiesdeltaprimeranksselmertwistsapplication
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

This paper studies the Selmer group of an abelian variety, a finite group built from local information that controls the rank of the rational points. For abelian varieties with complex multiplication (CM), the author considers all twists by characters of order p, obtained by tensoring with a one-dimensional character module. The Selmer group of each twist is the intersection of a fixed global cohomology group with a collection of local subspaces. Under the assumptions (H0)-(H3), the local subspaces are maximal isotropic in a symplectic or unitary quadratic space, depending on whether the prime p is ramified or inert in the CM field.

The author imports a Markov model from Klagsbrun, Mazur and Rubin to track how the dimension of these Selmer groups changes as more ramified primes are allowed. The transition probabilities depend only on the current dimension, and the invariant distribution is computed exactly. The main theorem says that, after ordering characters by a so-called fan structure that sorts them by ramification, the fraction of twists whose Selmer rank is r converges to D^epsilon_q(r).

The paper applies this to twisted Fermat curves X^p+Y^p=delta. The Jacobian of a related curve is a CM abelian variety, and a zero-Selmer condition forces the absence of rational points. Since the zero probability D^Sym_p(0) is larger than 1 minus 1/(p-1), the proportion of delta for which the equation has no solution is large, and becomes arbitrarily close to 1 as p grows.

Extended reading notes

Core claim

The central load-bearing assertion is Theorem 1.4: under hypotheses (CM), (H0)-(H3), and either F=F_{d-1} not equal F_d with d at least 2 or F_1=F_d, the p-Selmer ranks modulo torsion of the twists A_chi obey lim_k lim_X #{chi in Delta(k,X): dim_k(Sel_p(A_chi)/kappa_{A_chi,p}(A_chi(F)_tor))=r}/|Delta(k,X)| = D^epsilon_q(r). If true, this distribution result implies Theorem 1.1: the fan-structure proportion of delta in F^times/F^times p for which X^p+Y^p=delta has no nontrivial F-solution is at least D^Sym_p(0)=product_{i>=1}(1+p^{-i})^{-1}.

Load-bearing premise

The load-bearing premise that is most fragile is the imported local quadratic-space theorem from the companion paper [33, Proposition 4.4 and Theorem 5.4], cited as Theorem 4.1 here: for every place v, the local Tate pairing makes (H^1(F_v,T), h_v) a metabolic symplectic k-space (CMS) or unitary k-space (CMU). All later claims that local Kummer images are maximal isotropic subspaces, the counts of ramified isotropic lines in Propositions 4.2 and 4.11, and the Markov transition probabilities in Theorem 5.7 depend on this structure, which is not proved in this preprint and is only available in a companion article by the same author. If that structure theorem failed or required additional hypotheses, the central distribution theorem would not follow.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the p-Selmer ranks of p-th twists A_chi of a CM abelian variety A over a number field F. Under hypotheses (CM), (H0)-(H3) and either F=F_{d-1} != F_d with d>=2 or F_1=F_d, the main theorem (Theorem 1.4) asserts that, along a "fan-structure" stratification of the character group, the ranks of Sel_p(A_chi) modulo the Kummer image of torsion obey the symplectic or unitary distribution D_q^epsilon. The proof follows the Markov-model strategy of Klagsbrun-Mazur-Rubin: local Kummer images are maximal isotropic subspaces of quadratic cohomology spaces, and the rank transitions are governed by a Markov operator whose invariant distribution is D_q^epsilon. Applications include a lower bound for the proportion of delta in F^times/F^{times p} for which the twisted Fermat equation X^p+Y^p=delta has no nontrivial F-solution (Theorem 1.1), and results on rank growth in cyclic extensions (Theorems 1.6-1.8). The paper is structured and explicit about its dependence on the companion article [33] for the local quadratic-space structure.

Significance. If the companion result [33] is correct, this is a substantial contribution: it gives closed-form distributions for Selmer ranks in a large family of CM twists, with explicit numerical values and a concrete arithmetic application to twisted Fermat curves. The Markov architecture is clearly explained, no free parameters enter the distributions, and the paper is transparent about the main external input and about the omega-not-in-O obstruction. The applications to Fermat curves and cyclic extension rank growth are natural and nontrivial. The significance is, however, conditional on the cited quadratic-space structure theorem and on the sketched Diophantine passage in Theorem 1.1.

major comments (2)
  1. [§1.1, proof of Theorem 1.1] Theorem 4.1 is the load-bearing input that makes the local quadratic spaces metabolic symplectic or unitary, and it is only cited from [33, Proposition 4.4 and Theorem 5.4]; no proof is given in this manuscript. This structure is used immediately in Proposition 4.2 to identify unramified cohomology as maximal isotropic and to obtain |I_v^ram|=p, and that count propagates through Proposition 4.11, Proposition 4.14 and Theorem 5.7 into the Markov transition probabilities and hence into the limiting distribution in Theorem 7.9 and Corollary 7.10. In particular, the manuscript must clarify that Theorem 4.1 applies when omega is not in O: Section 1.4 notes that in this case A_chi generally does not admit a symmetric isogeny of degree prime to p with Rosati involution restricting to complex conjugation, and Proposition 4.3 uses a Bloch-Kato orthogonality argument for the maximal isotropy of L_{A,v,chi}. The counting of isotropic lines in Proposition 4.2(3), however, still depends directly on Theorem 4.1. If [33, Theorem 5.4] implicitly assumes the existence of such a polarization on the twisted variety, then the omega-not-in-O branch would not be covered by the cited theorem and the distribution theorem would not follow. Please add a proof, or a precise statement of the applicable hypotheses, or an explicit reduction of the omega-not-in-O case to the cited theorem.
  2. [§1.1, proof of Theorem 1.1] The passage from rank-zero Jacobians to nonsolubility of X^p+Y^p=delta is the arithmetic content of the Fermat application, but the manuscript only says "by the same argument using Northcott's finiteness theorem as in [9]". Since the setting here is an arbitrary number field containing mu_p, rather than the totally real fields of [9], a complete proof or a precise theorem quoted with its hypotheses is needed to justify that all but finitely many delta in Pi_1 give no nontrivial F-solution. This is a load-bearing step for Theorem 1.1 and Corollary 1.2.
minor comments (4)
  1. [§7, Eq. (7.1)] The displayed equation has a stray factor 8d(B,k) on the left-hand side; as printed the equality is false. The subsequent argument uses the identity without this factor, so this appears to be a typo, but it should be corrected.
  2. [§5, Eq. (5.2)] The word "otherwsie" should be "otherwise".
  3. [§1.4] "Rostai involution" should be "Rosati involution".
  4. [§1.2, after Definition 1.3] The word "fan-structue" should be "fan-structure", and in the Mathematics Subject Classification line "Primiary" should be "Primary".

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Selmer-rank distribution is obtained from local Kummer-image counts, effective Chebotarev, and the KMR Markov transfer; the companion-paper quadratic-space theorem is independent input, not a fitted or self-referential premise.

full rationale

The central theorem (Theorem 1.4) is not derived by fitting or by renaming an assumed distribution. The limiting distribution D_q^epsilon is the unique invariant distribution of the Markov operator M^epsilon (Proposition 5.9), and M^epsilon's transition probabilities come from Proposition 4.14, which counts how local Kummer images meet a fixed maximal isotropic subspace; those counts use Proposition 4.11 and Proposition 4.2. None of these steps take the target Selmer-rank distribution as an input. The main external input is Theorem 4.1, imported from the author's companion paper [33] as "a special case of [33, Proposition 4.4 and Theorem 5.4]". That is a structural theorem about local Tate pairings on H^1(F_v,T) being metabolic symplectic/unitary spaces; it does not assert the Selmer-rank distribution and is not fitted to the data whose distribution is predicted. The paper itself flags (Section 1.4) that for omega not in O no symmetric prime-to-p polarization exists on A_chi, forcing the Bloch-Kato argument in Proposition 4.3; this is a completeness or correctness risk about [33], not circularity. The score is 2 rather than 0 only because the load-bearing local-space input is cited to the author's own companion paper, so the proof is not fully self-contained; as an independent theorem with stated hypotheses not containing the target result, it is legitimate support rather than circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on hypotheses (H0)-(H3) plus imported theorems from [13] and [33]; no numerical free parameters are fitted. The only non-standard asserted input is the Northcott step for the Fermat application.

assumptions (6)
  • domain assumption Hypothesis (H2): T=A[p] is a simple F_p[G_F]-module.
    Used in Proposition 5.2 to force Gal(F_omega/F_1) to be isomorphic to a direct sum of copies of T, which underlies the Markov transition probabilities. The paper notes (H2) is automatic in the (CMS) case.
  • domain assumption Theorem 4.1 from [33]: local Tate pairings make H^1(F_v,T) a metabolic symplectic or unitary k-space.
    This is the foundation for all maximal-isotropic and quadratic-space counting; it is not proved in the present preprint.
  • domain assumption There exists a symmetric isogeny lambda:A to A^vee that is dagger-sesquilinear, with p not dividing deg(lambda).
    Assumed in the setup in Section 1.2 and needed for the local quadratic structure; Appendix A.2 constructs such lambda in special cases used for Theorem 1.8.
  • standard math Effective Chebotarev theorem with convergence rate L (Theorem 5.1).
    Imported from Serre and Lagarias-Odlyzko via [13, Theorem 8.1], used to control densities of places in the Markov and global passage arguments.
  • ad hoc to paper For all but finitely many delta in Pi_1, the twisted Fermat equation X^p+Y^p=delta has no nontrivial F-solution.
    Asserted in the proof of Theorem 1.1 via a Northcott finiteness argument with a citation to [9], but not proved in this paper.
  • domain assumption X(A_chi)[p^infinity] is finite for all chi in C(F) in Theorem 1.8.
    Stated as an assumption before Theorem 1.8 and used to relate parity of Selmer ranks to parity of Mordell-Weil ranks via [33, Corollary 6.4-6.5].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Selmer ranks in twists of CM abelian varieties." pith.science (2026). https://pith.science/paper/M5UMQQ6B

@misc{pith2026250421274,
  author       = {Pith},
  title        = {Pith review of: Selmer ranks in twists of CM abelian varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5UMQQ6B}},
  note         = {Machine review of arXiv:2504.21274}
}
abstract

We prove the Selmer ranks in certain families of $p$-th twists of CM abelian varieties obey the symplectic or unitary distributions. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curves $X^p+Y^p=\delta$ over a number field containing a primitive $p$-th root of unity are ``largely" unsolvable as $\delta$ varies. We also discuss the rank growth in cyclic extensions of prime degree for CM abelian varieties.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quadratic spaces and Selmer groups of abelian varieties with multiplication

    math.NT 2025-04 accept novelty 7.0 of 10

    For abelian varieties over global fields with multiplication by an order, the Selmer group is the intersection of two maximal isotropic subspaces in an orthogonal, symplectic, unitary, or split unitary quadratic space.

Reference graph

Works this paper leans on

36 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [33]

    J. Shu. Quadratic spaces and Selmer groups of abelian varieties with multiplication. arXiv:2504.21272, 2025

    work page Pith review arXiv 2025

  2. [9]

    Diaconu and Y

    A. Diaconu and Y. Tian. Twisted Fermat curves over totally real fields. Ann. of Math. , 162:1353–1376, 2005

    work page 2005

  3. [1]

    Alp¨ oge, M

    L. Alp¨ oge, M. Bhargava, and A. Shnidman. Integers expressible as the sum of two rational cubes. arXiv:2210.10730., 2022

    arXiv 2022

  4. [2]

    Amir-Khosravi

    Z. Amir-Khosravi. Serre’s tensor construction and moduli of abelian schemes. Manuscripta Math., 156:409–456, 2018

    work page 2018

  5. [3]

    Bloch and K

    S. Bloch and K. Kato. L-functions and Tamagawa numbers of motives. In P. Cartier and et al., editors, The Grothendieck Festschrift I, volume 86 of Progr. Math., pages 333–400. Birkhauser Boston, 1990

    work page 1990

  6. [4]

    Bourdon and P

    A. Bourdon and P. L. Clark. Torsion points and Galois representations on CM elliptic curves. Pacific J. Math. , 305(1):43–88, 2020. 33

    work page 2020

  7. [5]

    Cube sum problem and an explicit Gross-Zagier formula

    Li Cai, Jie Shu, and Ye Tian. Cube sum problem and an explicit Gross-Zagier formula. Amer. J. Math. , 139(3):785–816, 2017

    work page 2017

  8. [6]

    B. Conrad. Gross-Zagier revisited. In Heegner points and Rankin L-series , volume 49 of Math. Sci. Res. Inst. Publ., pages 67–163. Cambridge University Press, 2004

    work page 2004

Show all 36 references
  1. [7]

    C. Cornut. Mazur’s conjecture on higher Heegner points. Invent. Math., 148(3):301–314, 2002

    2002

  2. [8]

    David, J

    C. David, J. Fearnley, and H. Kisilevsky. Vanishing of L-functions of elliptic curves over number fields. In Ranks of elliptic curves and random matrix theory , volume 341 of London Math. Soc. Lecture Notes , pages 247–259. Cambridge, 2007

    2007

  3. [10]

    Edixhoven, G

    B. Edixhoven, G. van der Geer, and B. Moonen. Abelian varieties . In preparation. Available at https://gerard.vdgeer.net/AV.pdf

  4. [11]

    B. H. Gross and J. Parson. On the local divisibility of Heegner points. Number theory, analysis and geometry: In Memory of Serge Lang , pages 215–241, 2012

    2012

  5. [12]

    E. W. Howe. Isogeny classes of abelian varieties with no principal polarizations. In F. Oort C. Faber, G. van der Geer, editor, Moduli of abelian varieties , volume 195 of Progr. Math. Birkhauser, Basel, 2001

    2001

  6. [13]

    Klagsbrun, B

    Z. Klagsbrun, B. Mazur, and K. Rubin. A Markov model for Selmer ranks in families of twists. Compositio Mathematica, 150(7):1077–1106., 2014

    2014

  7. [14]

    Koymans and A

    P. Koymans and A. Smith. Sums of rational cubes and the 3-Selmer group. arXiv:2405.09311, 2024

    2024 arXiv

  8. [15]

    Lagarias and A.M

    J.C. Lagarias and A.M. Odlyzko. Effective versions of the chebotarev density theorem. In A. Fr¨ ohlich, editor, Algeraic Number Fields: L-functions and Galois properties . Academic Press, 1977

    1977

  9. [16]

    D. Lieman. Nonvanishing of L-series associated to cubic twists of elliptic curves. Ann. of Math. , 140:81–108, 1994

    1994

  10. [17]

    B. Mazur. Modular curves and arithmetic. In Proc. Inter. Congr. Math. , Warszawa, 1983

    1983

  11. [18]

    Mazur and K

    B. Mazur and K. Rubin. Finding large Selmer rank via an arithmetic theory of local constants. Ann. of Math. , 166:579–612, 1994

    1994

  12. [19]

    Mazur and K

    B. Mazur and K. Rubin. Arithmetic conjectures suggested by statistical behavior of modular symbols. Exp. Math., 32(4):657–672, 2023

    2023

  13. [20]

    Mazur, K

    B. Mazur, K. Rubin, and M. Larsen. Diophantine stability. Amer. J. Math. , 140(3):571–616, 2018

    2018

  14. [21]

    Mazur, K

    B. Mazur, K. Rubin, and A. Silverberg. Twisting commutative algebraic groups. Journal of Algebra, 314:419–438, 2007

    2007

  15. [22]

    J. S. Milne. Arithmetic duality theorems. Academic Press Inc Boston Ma, 2006

    2006

  16. [23]

    Neukirch

    J. Neukirch. Algeraic Number Theory. Springer Berlin, Heidelberg, 1992

    1992

  17. [24]

    J. R. Norris. Markov chains. Cambridge University Press, 1997

    1997

  18. [25]

    K. Rubin. Euler systems and Kolyvagin systems. IAS/Park City Mathematics Series , XX(XXXX)

  19. [26]

    J.-P. Serre. Quelques applications du th´ eor` eme de chebotarev.Publ. Math. Inst. Hautes ´Etudes Sci., 54:123–201, 1981

    1981

  20. [27]

    J.-P. Serre. Cohomologie Galisienne , volume 5 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 5th edition, 1994

    1994

  21. [28]

    J.-P. Serre. Local fields, volume 67 of Graduate Texts in Mathematics . Springer New York, 1995

    1995

  22. [29]

    Serre and J

    J.-P. Serre and J. Tate. Good reduction of abelian varieties. Ann. of Math. , 68:492–517, 1968

    1968

  23. [30]

    G. Shimura. Abelian varieties with complex multiplication and modular functions . Princeton University Press, Princeton, NJ, 1996

    1996

  24. [31]

    J. Shu. Root number for the Jacobian varieties of Fermat curves. Journal of Number Theory, 226:243–270, 2021

    2021

  25. [32]

    J. Shu. Selmer groups and the parity conjectures for Jacobian varieties of Fermat curves. preprint, 2021

    2021

  26. [34]

    J. Tate. Duality theorems in Galois cohomology over number fields. In Proc. Inter. Congr. Math., pages 234–241, Stockholm, 1962

    1962

  27. [35]

    V. Vatsal. Special values of anticyclotomic L-functions. Duke Math. J. , 116(2):219–261, 2003

    2003

  28. [36]

    A. J. Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. , 141:443–551, 1995. School of Mathematical Sciences, Tongji University, Shanghai 200092, P. R. China Email address: shujie@tongji.edu.cn 34

    1995

Pith tools

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