pith. sign in

arxiv: 2604.05739 · v1 · submitted 2026-04-07 · 🧮 math.NT

On Iwasawa theory of abelian varieties over mathbb{Z}_p²-extension with applications to Diophantine stability and integally Diophantine extensions

Meng Fai Lim This is my paper

Pith reviewed 2026-05-10 18:21 UTC · model grok-4.3

classification 🧮 math.NT
keywords Iwasawa theoryabelian varietiesZ_p^2-extensionDiophantine stabilityMazur growth conjectureordinary reductionsupersingular reduction
0
0 comments X

The pith

Iwasawa theory for abelian varieties with potentially good ordinary reduction yields Diophantine stability over Z_p^2-extensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes results in Iwasawa theory for abelian varieties that possess potentially good ordinary reduction at all primes above p, working specifically over Z_p^2-extensions of the base field. These results are applied directly to questions of Diophantine stability and to the identification of integally Diophantine extensions. Additional consequences refine statements of the Mazur growth conjecture. The same framework is carried over to elliptic curves that instead have good supersingular reduction at p. A sympathetic reader cares because the work connects the structure of Selmer groups in infinite p-adic towers to the stability of rational points and integral models in those towers.

Core claim

We present certain results on the Iwasawa theory of an abelian variety with potentially good ordinary reduction at all primes above p. These are then applied to study Diophantine stability and integally Diophantine extensions. Along the way, we also obtain some results pertaining to Mazur growth conjecture which refine previous results. Finally, we extend our investigation to the case of an elliptic curve with good supersingular reduction at the prime p and make a similar analysis.

What carries the argument

Iwasawa-theoretic control theorems for Selmer groups of abelian varieties over Z_p^2-extensions

If this is right

  • Diophantine stability holds for the abelian varieties in the Z_p^2-extension.
  • Integally Diophantine extensions of the base field are identified via the Iwasawa data.
  • The Mazur growth conjecture receives refinements that improve earlier bounds.
  • Analogous stability conclusions apply to elliptic curves with good supersingular reduction at p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The control theorems might extend to Z_p^d-extensions for d greater than 2 if analogous Selmer bounds can be proved.
  • Explicit computations for elliptic curves with ordinary reduction at p could test the stability predictions in low layers of the tower.
  • The framework may inform questions about integral points on higher-dimensional varieties inside the same infinite extensions.

Load-bearing premise

The abelian variety has potentially good ordinary reduction at all primes above p (or good supersingular reduction in the elliptic curve case), without which the control theorems do not apply.

What would settle it

An explicit abelian variety satisfying the reduction hypothesis for which the Selmer rank grows unboundedly or violates the refined growth prediction inside a concrete Z_p^2-extension.

read the original abstract

We present certain results on the Iwasawa theory of an abelian variety with potentially good ordinary reduction at all primes above $p$. These are then applied to study Diophantine stability and integally Diophantine extensions. Along the way, we also obtain some results pertaining to Mazur growth conjecture which refine previous results of Gajek-Leonard, Hatley, Kundu and Lei. Finally, we extend our investigation to the case of an elliptic curve with good supersingular reduction at the prime $p$ and make a similar analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes Iwasawa-theoretic control theorems for abelian varieties possessing potentially good ordinary reduction at all primes above p, over the two-variable Z_p^2-extension. These theorems are applied to questions of Diophantine stability and the existence of integally Diophantine extensions. The paper also derives refinements of earlier results on the Mazur growth conjecture due to Gajek-Leonard, Hatley, Kundu and Lei, and concludes with an analogous analysis in the case of elliptic curves with good supersingular reduction at p.

Significance. If the control theorems and their applications are valid, the work advances the study of Iwasawa modules over multi-variable p-adic Lie extensions and supplies concrete links to Diophantine geometry. The explicit refinements to the Mazur growth conjecture and the separate treatment of the supersingular case constitute incremental but useful progress beyond the cited literature.

minor comments (2)
  1. The introduction would benefit from a brief roadmap indicating which theorems are new versus which are refinements of the cited works of Gajek-Leonard et al.
  2. Notation for the Iwasawa algebra and the associated Lambda-modules should be fixed consistently across sections; occasional shifts between Lambda and the completed group ring are distracting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the detailed summary, and the recommendation to accept. We appreciate the recognition of the control theorems, their applications to Diophantine stability, the refinements to the Mazur growth conjecture, and the treatment of the supersingular case.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states its main results (Iwasawa control theorems for abelian varieties with potentially good ordinary reduction, applications to Diophantine stability, and refinements of the Mazur growth conjecture) under explicit technical hypotheses drawn from prior literature by other authors. The derivations rely on standard Lambda-module structures and control theorems that are not redefined or fitted within the paper itself; the supersingular case is handled separately with analogous assumptions. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The argument chain is self-contained against external benchmarks in the Iwasawa theory literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5393 in / 1277 out tokens · 83254 ms · 2026-05-10T18:21:55.126112+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    D. Bump, S. Friedberg and J. Hoffstein, Nonvanishing forL-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543-618

    work page 1990

  2. [2]

    Coates and R

    J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), no. 1-3, 129-174

    work page 1996

  3. [3]

    Denef and L

    J. Denef and L. Lipshitz, Diophantine sets over some rings of algebraic integers, J. Lond. Math. Soc. (2) 18 (1978), no. 3, 385-391

    work page 1978

  4. [4]

    Dokchitser and V

    T. Dokchitser and V. Dokchitser, Computations in non-commutative Iwasawa theory. With an appendix by J. Coates and R. Sujatha, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 211-272

    work page 2007

  5. [5]

    Gajek-Leonard, J

    R. Gajek-Leonard, J. Hatley, D. Kundu and A. Lei, On a conjecture of Mazur predicting the growth of Mordell-Weil ranks inZ p-extensions, Math. Res. Lett. 32 (2026), no. 6, 1877-1912

    work page 2026

  6. [6]

    Garcia-Fritz and H

    N. Garcia-Fritz and H. Pasten, Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points, Math. Ann. 377 (2020), no. 3-4, 989-1013

    work page 2020

  7. [7]

    Greenberg, Iwasawa theory forp-adic representations, in: Algebraic Number Theory-in honor of K

    R. Greenberg, Iwasawa theory forp-adic representations, in: Algebraic Number Theory-in honor of K. Iwasawa, ed. J. Coates, R. Greenberg, B. Mazur and I. Satake, Adv. Std. in Pure Math. 17, 1989, pp. 97-137

    work page 1989

  8. [8]

    Greenberg, Iwasawa theory for elliptic curves, in: Arithmetic theory of elliptic curves (Cetraro, 1997), 51-144, Lecture Notes in Math., 1716, Springer, Berlin, 1999

    R. Greenberg, Iwasawa theory for elliptic curves, in: Arithmetic theory of elliptic curves (Cetraro, 1997), 51-144, Lecture Notes in Math., 1716, Springer, Berlin, 1999

    work page 1997

  9. [9]

    Greenberg, Introduction to Iwasawa theory for elliptic curves, in: Arithmetic Algebraic Geometry, IAS/Park City Mathematics Series, 9 (eds

    R. Greenberg, Introduction to Iwasawa theory for elliptic curves, in: Arithmetic Algebraic Geometry, IAS/Park City Mathematics Series, 9 (eds. B. Conrad and K. Rubin) (American Mathematical Society, Providence, RI, 1999), 407-464

    work page 1999

  10. [10]

    B. H. Gross and D. Zagier, Heegner points and derivatives ofL-series, Invent. Math. 84 (1986), no. 2, 225-320

    work page 1986

  11. [11]

    Hachimori and K

    Y. Hachimori and K. Matsuno, An analgoue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), no. 3, 581-601

    work page 1999

  12. [12]

    Hachimori and T

    Y. Hachimori and T. Ochiai, Notes on non-commutative Iwasawa theory, Asian J. Math. 14 (2010), No. 1, 11-18

    work page 2010

  13. [13]

    Imai, A remark on the rational points of abelian varieties with values in cyclotomicZ p-extensions, Proc

    H. Imai, A remark on the rational points of abelian varieties with values in cyclotomicZ p-extensions, Proc. Japan Acad. 51 (1975), 12-16

    work page 1975

  14. [14]

    Iovita and R

    A. Iovita and R. Pollack, Iwasawa theory of elliptic curves at supersingular primes overZ p-extensions of number fields, J. reine angew. Math. 598 (2006), 71-103

    work page 2006

  15. [15]

    Jannsen, A spectral sequence for Iwasawa adjoints, M¨ unster J

    U. Jannsen, A spectral sequence for Iwasawa adjoints, M¨ unster J. Math. 7 (2014), no. 1, 135-148

    work page 2014

  16. [16]

    Kato,p-adic Hodge theory and values of zeta functions of modular forms, in: Cohomologiesp-adiques et applications arithm´ etiques

    K. Kato,p-adic Hodge theory and values of zeta functions of modular forms, in: Cohomologiesp-adiques et applications arithm´ etiques. III., Ast´ erisque 295, 2004, ix, pp. 117-290

    work page 2004

  17. [17]

    B. D. Kim, Signed-Selmer groups over theZ 2 p-extension of an imaginary quadratic field, Canad. J. Math. 66 (2014), no. 4, 826-843

    work page 2014

  18. [18]

    Kleine, A

    S. Kleine, A. Matar and R. Sujatha, On theM H(G)-property, Math. Proc. Cambridge Philos. Soc. 179 (2025), no. 2, 449-501. 21

    work page 2025

  19. [19]

    Kleine, A

    S. Kleine, A. Matar and R. Sujatha, On theM H(G)-property for Selmer groups at supersingular reduction, arXiv:2601.08612 [mathNT]

    work page arXiv

  20. [20]

    Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent

    S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1-36

    work page 2003

  21. [21]

    [KP25] Peter Koymans and Carlo Pagano

    P. Koymans and C. Pagano, Hilbert’s tenth problem via additive combinatorics, arXiv:2412.01768v3 [math.NT]

    work page arXiv

  22. [22]

    Kundu and A

    D. Kundu and A. Lei, Generalized Mazur’s growth number conjecture, to appear in Bull. Aust. Math. Soc

    work page

  23. [23]

    Kundu and A

    D. Kundu and A. Lei, Mazur’s growth number conjecture in the rank one case, to appear in Quart. J. Math

    work page

  24. [24]

    Kundu, A

    D. Kundu, A. Lei and F. Sprung, Studying Hilbert’s 10th problem via explicit elliptic curves, Math. Ann. 390 (2024), no. 4, 5153-5183

    work page 2024

  25. [25]

    Lei and M

    A. Lei and M. F. Lim. Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes abovep, J. Th´ eor. Nombres Bordeaux 33 (2021), no. 3.2, 997-1019

    work page 2021

  26. [26]

    Lei and B

    A. Lei and B. Palvannan, Codimension two cycles in Iwasawa theory and ellpitic curves with supserinsgular reduction, Forum Math. Sigma 7 (2019), Paper No. e2, 81

    work page 2019

  27. [27]

    M. F. Lim, On order of vanishing of characteristic elements, Forum Math. 34 (2022), No. 4, 1051-1080

    work page 2022

  28. [28]

    M. F. Lim, Structure of fine Selmer groups overZ p-extensions, Math. Proc. Cambridge Philos. Soc. 176 (2024), no. 2, 287-308

    work page 2024

  29. [29]

    Longo and S

    M. Longo and S. Vigni, Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes, Boll. Unione Mat. Ital. 12 (2019), no. 3, 315-347

    work page 2019

  30. [30]

    Mazur, Rational points of abelian varieties with values in towers of number fields, Invent

    B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972) 183-266

    work page 1972

  31. [31]

    Mazur, Modular curves and arithmetic

    B. Mazur, Modular curves and arithmetic. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, pp. 185-211 (1984)

    work page 1983

  32. [32]

    Mazur and K

    B. Mazur and K. Rubin and M. Larsen, Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541-575

    work page 2010

  33. [33]

    Mazur, K

    B. Mazur, K. Rubin and M. Larsen, Diophantine stability, Am. J. Math. 140 (2018), No. 3, 571-616

    work page 2018

  34. [34]

    Mazur, K

    B. Mazur, K. Rubin and A. Shlapentokh, Existential definability and diophatine stability, J. Number Theory 254 (2024), 1-64

    work page 2024

  35. [35]

    M. R. Murty and V. K. Murty, Mean values of derivatives of modularL-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475

    work page 1991

  36. [36]

    V. K. Murty and Y. Ouyang, The growth of Selmer ranks of an Abelian variety with complex multiplication, Pure. Appl. Math. Q. 2 (2026), no. 2, 539-555

    work page 2026

  37. [37]

    Neukirch, A

    J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd edn., Grundlehren Math. Wiss. 323 (Springer-Verlag, Berlin, 2008)

    work page 2008

  38. [38]

    Nekov´ aˇ r, Growth of Selmer groups of Hilbert modular forms over ring class fields, Ann

    J. Nekov´ aˇ r, Growth of Selmer groups of Hilbert modular forms over ring class fields, Ann. Sci.´Ec. Norm. Sup´ er. (4) 41(6) (2008), 1003-1022

    work page 2008

  39. [39]

    Nekov´ aˇ r, Selmer complexes, Ast´ erisque 310, 2006

    J. Nekov´ aˇ r, Selmer complexes, Ast´ erisque 310, 2006

    work page 2006

  40. [40]

    Ochi and O

    Y. Ochi and O. Venjakob, On the ranks of Iwasawa modules overp-adic Lie extensions, Math. Proc. Camb. Phil. Soc. 135 (2003), 25-43

    work page 2003

  41. [41]

    Perrin-Riou, Arithm´ etique des courbes elliptiques et th´ eorie d’Iwasawa, M´ em

    B. Perrin-Riou, Arithm´ etique des courbes elliptiques et th´ eorie d’Iwasawa, M´ em. Soc. Math. France (N.S.) No. 17 (1984), 130 pp

    work page 1984

  42. [42]

    Ray, Remarks on Hilbert’s tenth problem and the Iwasawa theory of ellpitic curves, Bull

    A. Ray, Remarks on Hilbert’s tenth problem and the Iwasawa theory of ellpitic curves, Bull. Aust. Math. Soc. 107 (2023), no. 3, 440-450

    work page 2023

  43. [43]

    Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent

    K. Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math. 93 (1988), 701-713

    work page 1988

  44. [44]

    Shlapentokh, Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers, Trans

    A. Shlapentokh, Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers, Trans. Am. Math. Soc. 360 (2008), no. 7, 3541-3555

    work page 2008

  45. [45]

    Wingberg, On the rational points of abelian varieties overZ p-extensions of number fields, Math

    K. Wingberg, On the rational points of abelian varieties overZ p-extensions of number fields, Math. Ann. 279 (1987) 9-24. 22

    work page 1987