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Generalised class groups in dihedral and fake ℤp-extensions

  • Filippo A. E. Nuccio Mortarino Majno Di Capriglio EMAIL logo
Veröffentlicht/Copyright: 9. Juni 2025
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 75 Heft 3

Abstract

We study generalised class groups in dihedral extensions of number fields, focusing both on its p-adic and ℓ-adic properties along fake ℤp-extensions, where p and ℓ are distinct odd primes. Our results generalise previous works in the (pro)-dihedral setting, with applications to the study of more general fake Iwasawa theories beyond the classical case of standard class groups.

Keywords: Iwasawa theory; fake ℤp-extension; generalised class group; dihedral extensions; growth of class numbers
MSC 2010: Primary 11R29; 11R20; Secondary 11R34; 11R37

This work is licensed under a CC-BY 4.0 licence. I am grateful to the two referees whose comments and suggestions helped improved the presentation. I am also deeply indebted to Luca Caputo for his comments on a first draft of the manuscript, in particular for pointing out an error in the first version of Proposition 2.11.

I take this opportunity to correct two typos introduced by the copy-editor while preparing the publisher version of my joint work [CN20]:

  • p. 5, line -5: . . . cδ is multiplied by −1 on H2(G, ℤ)… should be replaced with . . . cδ is multiplication by −1 on H2(G, ℤ)…

  • p. 11, line 1: . . . since NL/K ∘ι L/K is multiplied by 2 = [L : K] … should be replaced with . . . since NL/K ∘ι L/K is multiplication by 2 = [L : K] …

  1. (Communicated by István Gaál)

References

[ABK+24] Asgeirson, D.—Brasca, R.—Kuhn, N.—Nuccio Mortarino Majno Di Capriglio, F. A. E.—Topaz, A.: Categorical foundations of formalized condensed mathematics, J. Symb. Log. (2024), 1–28.10.1017/jsl.2024.69Suche in Google Scholar

[BDNN21] Baanen, A.—Dahmen, S. R.—Narayanan, A.—Nuccio Mortarino Majno Di Capriglio, F. A. E.: A formalization of Dedekind domains and class groups of global fields. In: 12th International Conference on Interactive Theorem Proving (ITP 2021), Dagstuhl, Germany (L. Cohen, C. Kaliszyk, eds.), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 193, Schloss Dagstuhl–Leibniz-Zentrum f¨ur Informatik, 2021, pp. 5:1–5:19.Suche in Google Scholar

[BDNN22] Baanen, A.—Dahmen, S. R.—Narayanan, A.—Nuccio Mortarino Majno Di Capriglio, F. A. E.: A formalization of Dedekind domains and class groups of global fields, J. Automat. Reason. 66(4) (2022), 611–637.10.1007/s10817-022-09644-0Suche in Google Scholar

[BL23] Bandini, A.—Longhi, I.: The algebraZℓZpd and applications to Iwasawa theory, https://arxiv.org/abs/2312.04666v1 (2023), version 1.Suche in Google Scholar

[BN18] Billerey, N.—Nuccio Mortarino Majno Di Capriglio, F. A. E.: Représentations galoisi-ennes diédrales et formes à multiplication complexe, J. Théor. Nombres Bordeaux 30(2) (2018), 651–670.10.5802/jtnb.1043Suche in Google Scholar

[CN20] Caputo, L.—Nuccio Mortarino Majno Di Capriglio, F. A. E.: Class number formula for dihedral extensions, Glasgow Math. J. 62(2) (2020), 323–353.10.1017/S0017089519000144Suche in Google Scholar

[CN23] Caputo, L.—Nuccio Mortarino Majno Di Capriglio, F. A. E.: Cohomology of normic systems and fake ℤp-extensions, New York J. Math. 29 (2023), 1196–1272.Suche in Google Scholar

[CS10] Coates, J.—Sujatha, R.: Galois Cohomology of Elliptic Curves, 2nd ed., Narosa Publishing House, New Delhi, Tata Institute of Fundamental Research Publications, Mumbai, 2010.Suche in Google Scholar

[dFF22] De Frutos-FernáNdez, M. I.: Formalizing the Ring of Adèles of a Global Field. In: 13th International Conference on Interactive Theorem Proving (ITP 2022), Dagstuhl, Germany, (J. Andronick, L. de Moura, eds.), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 237, Schloss Dagstuhl– Leibniz-Zentrum f¨ur Informatik, 2022, pp. 14:1–14:18.Suche in Google Scholar

[dFFN24] De Frutos-FernáNdez, M. I.—Nuccio Mortarino Majno Di Capriglio, F. A. E.: A formalization of complete discrete valuation rings and local fields, Proceedings of the 13th ACM SIG-PLAN International Conference on Certified Programs and Proofs, New York, NY, USA, CPP 2024, Association for Computing Machinery, 2024, pp. 190–204.10.1145/3636501.3636942Suche in Google Scholar

[Iwa73] Iwasawa, K. On ℤl-extensions of algebraic number fields, Ann. Math. (1973), 246–326.10.2307/1970784Suche in Google Scholar

[KL23] Kundu, D.—Lei, A.: Growth of p-parts of ideal class groups and fine Selmer groups in ℤq-extensions with p = q, Acta Arith. 207(4) (2023), 297–313.10.4064/aa220518-28-2Suche in Google Scholar

[KNS24] Kundu, D.—Nuccio Mortarino Majno Di Capriglio, F. A. E.—Sujatha, R.: Structure of fine Selmer groups in abelian p-adic Lie extensions, Osaka J. Math. 61(1) (2024), 121–146.Suche in Google Scholar

[Liv23] Livingston, A.: Group Cohomology in the Lean Community Library. In: 14th International Conference on Interactive Theorem Proving (ITP 2023), Dagstuhl, Germany, (A. Naumowicz, R. Thiemann, eds.), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 268, Schloss Dagstuhl–Leibniz-Zentrum f¨ur Informatik, 2023, pp. 22:1–22:17.Suche in Google Scholar

[mC20] The Mathlib Community: The Lean mathematical library. In: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, POPL ’20, ACM, January 2020.10.1145/3372885.3373824Suche in Google Scholar

[NOR23] Nuccio Mortarino Majno Di Capriglio, F. A. E.—Ochiai, T.—Ray, J.: A formal model of Coleman families and applications to Iwasawa invariants, Ann. Math. Qué. 48 (2024), 453–475.10.1007/s40316-023-00217-0Suche in Google Scholar

[NS23] Nuccio Mortarino Majno Di Capriglio, F. A. E.—Sujatha, R.: Residual supersingular Iwasawa theory and signed Iwasawa invariants, Rend. Semin. Mat. Univ. Padova 149 (2023), 83–129.10.4171/rsmup/111Suche in Google Scholar

[NSW08] Neukirch, J.—Schmidt, A.—Wingberg, K.: Cohomology of Number Fields, 2nd ed., Grundlehren Math. Wiss., Vol. 323, Springer-Verlag, Berlin, 2008.10.1007/978-3-540-37889-1Suche in Google Scholar

[Nuc10] Nuccio Mortarino Majno Di Capriglio, F. A. E.: Cyclotomic units and class groups in ℤp-extensions of real abelian fields, Math. Proc. Cambridge Philos. Soc. 148(1) (2010), 93–106.10.1017/S0305004109990119Suche in Google Scholar

[Oza95] Ozaki, M: A note on the capitulation in ℤp-extensions, Proc. Japan Acad. Ser. A Math. Sci. 71(9) (1995), 218–219.10.3792/pjaa.71.218Suche in Google Scholar

[Rio25] Riou, J.: Formalization of derived categories in Lean/mathlib, Annals of Formalized Mathematics (2025), to appear; https://hal.science/hal-04546712v4Suche in Google Scholar

[Rub88] Rubin, K: On the main cojecture of Iwasawa theory for imaginary quadratic fields, Invent. Math. 93(3) (1988), 701–714.10.1007/BF01410205Suche in Google Scholar

[Ser67] Serre, J.-P.: Local class field theory. In: Algebraic Number Theory, Proc. Instructional Conf., Brighton, 1965, Academic Press, London, 1967, pp. 128–161.Suche in Google Scholar

[Tat67] Tate, J.: Global class field theory. In: Algebraic Number Theory, Proc. Instructional Conf., Brighton, 1965, Thompson, Washington, D. C., 1967, pp. 162–203.Suche in Google Scholar

[Wal77] Walter, C. D.: A class number relation in Frobenius extensions of number fields, Mathematika 24(2) (1977), 216–225.10.1112/S0025579300009128Suche in Google Scholar
Received: 2024-09-26
Accepted: 2025-03-06
Published Online: 2025-06-09
Published in Print: 2025-06-26

© 2025 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Joins of normal matrices, their spectrum, and applications
  2. On isbell’s density theorem for bitopological pointfree spaces II
  3. Every positive integer is a sum of at most n + 2 centered n-gonal numbers
  4. A generalisation of q-additive functions
  5. Generalised class groups in dihedral and fake ℤp-extensions
  6. Singular value bounds with applications to norm and numerical radius inequalities
  7. Coefficient bounds for convex functions associated with cosine function
  8. On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
  9. Numerical implementation of solving a boundary value problem including both delay and parameter
  10. Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
  11. Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
  12. On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
  13. A note on derivations into annihilators of the ideals of banach algebras
  14. Characterization of nonlinear mixed skew lie and jordan n-Type derivation on ∗-Algebras
  15. Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
  16. A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
  17. Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
https://doi.org/10.1515/ms-2025-0039
Schlagwörter für diesen Artikel
Iwasawa theory; fake ℤp-extension; generalised class group; dihedral extensions; growth of class numbers

Artikel in diesem Heft

  1. Joins of normal matrices, their spectrum, and applications
  2. On isbell’s density theorem for bitopological pointfree spaces II
  3. Every positive integer is a sum of at most n + 2 centered n-gonal numbers
  4. A generalisation of q-additive functions
  5. Generalised class groups in dihedral and fake ℤp-extensions
  6. Singular value bounds with applications to norm and numerical radius inequalities
  7. Coefficient bounds for convex functions associated with cosine function
  8. On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
  9. Numerical implementation of solving a boundary value problem including both delay and parameter
  10. Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
  11. Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
  12. On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
  13. A note on derivations into annihilators of the ideals of banach algebras
  14. Characterization of nonlinear mixed skew lie and jordan n-Type derivation on ∗-Algebras
  15. Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
  16. A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
  17. Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
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