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On the $\mathfrak{M}_H(G)$-property

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  24 June 2025

SÖREN KLEINE ,
AHMED MATAR  and
SUJATHA RAMDORAI
SÖREN KLEINE
Affiliation:
Institut für Anwendungssicherheit, Fakultät für Informatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85579 Neubiberg, Germany. e-mail: soeren.kleine@unibw.de
AHMED MATAR
Affiliation:
Department of Mathematics, University of Bahrain, P.O. Box 32038, Sukhair, Bahrain. e-mail: amatar@uob.edu.bh
SUJATHA RAMDORAI
Affiliation:
Department of Mathematics, 1984, Mathematics Road, University of British Columbia, Vancouver, V6T1Z2, Canada. e-mail: sujatha@math.ubc.ca
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Abstract

Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p. Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property for ${{\mathbb{Z}}}_p$-extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$-extension $K_\infty/K$. The equivalent conditions involve the growth of $\mu$-invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$-extensions in $K_\infty$, and the boundedness of $\lambda$-invariants as one runs over ${{\mathbb{Z}}}_p$-extensions of K inside of $K_\infty$.

Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$-extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$-property sometimes can be shifted to a larger base field K′.

MSC classification

Primary: 11R23: Iwasawa theory

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 53
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

Dedicated to the memory of John H. Coates

References

Atiyah, M. and Macdonald, I.. Introduction to Commutative Algebra (Adison-Wesley, 1969).Google Scholar
Brink, D.. Prime decomposition in the anti-cyclotomic extensions, Math. Comp., 76(260) (2007), 2127-2138.CrossRefGoogle Scholar
Bertolini, M. and Darmon, H.. Kolyvagin’s descent and Mordell–Weil groups over ring class fields, J. Reine Angew. Math. 412 (1990), 63–74.Google Scholar
Bloom, J. and Gerth III, F.. The Iwasawa invariant $\mu$ in the composite of two ${{\mathbb{Z}_p}}$ -extensions, J. Number Theory 13 (1981), 262–267.CrossRefGoogle Scholar
Bourbaki, N.. Commutative Algebra, (Hermann, Paris, 1972).Google Scholar
Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O.. The $GL_2$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. (2005), no. 101, 163–208.CrossRefGoogle Scholar
Coates, J. and Greenberg, R.. Kummer theory for Abelian varieties over local fields, Invent. Math. 124 (1996), 129–174.CrossRefGoogle Scholar
Coates, J., Schneider, P. and Sujatha, R.. Links between cyclotomic and $\text{GL}_2$ Iwasawa theory, Doc. Math. (2003), extra vol., 187–215.Google Scholar
Coates, J. and Sujatha, R.. Galois Cohomology of Elliptic Curves, 2nd. ed., Tata Institute of Fundamental Research Lectures on Mathematics, 88 (Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2010)Google Scholar
Coates, J. and Sujatha, R. On the $\mathfrak{M}_H(G)$ -conjecture, Non-abelian Fundamental Groups and Iwasawa Theory. London Math. Soc. Lecture Note Ser. 393 (Cambridge University Press, Cambridge, 2012). 132–161.CrossRefGoogle Scholar
Cremona, J. E.. Algorithms for modular elliptic curves, 2nd ed. (Cambridge University Press, Cambridge, 1997).Google Scholar
Cuoco, A.. The growth of Iwasawa invariants in a family. Compos. Math. 41(3) (1980), 415–437.Google Scholar
Cuoco, A., Monsky, P.. Class numbers in ${{\mathbb{Z}}}_p^d$ -extensions , Math. Ann. 255 (1981), 235–258.CrossRefGoogle Scholar
Cuoco, A.. Relations between invariants in ${{\mathbb{Z}}}_p^2$ -extensions, Math. Z. 181(2) (1982), 197–200.CrossRefGoogle Scholar
Cuoco, A.. Generalized Iwasawa invariants in a family, Compos. Math. 51(1) (1984), 89–103.Google Scholar
Dokchitser, T., Dokchitser, V.. On the Birch-Swinnerton-Dyer quotients modulo squares, Ann. of Math. 172(1) (2010), 567–596.CrossRefGoogle Scholar
Fukaya, T., Kato, K.. A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory. Proc. St. Petersburg Math. Soc. Vol. XII, Amer. Math. Soc. Transl. Ser. 2 219 (Amer. Math. Soc., 2006), 1--85.CrossRefGoogle Scholar
Greenberg, R.. The Iwasawa invariants of $\Gamma$ -extensions of a fixed number field, Amer. J. Math. 95 (1973), 204–214.CrossRefGoogle Scholar
Greenberg, R.. Iwasawa theory for elliptic curves. Lecture Notes in Math. 1716 (Springer, New York 1999), pp.51-144.Google Scholar
Greenberg, R.. Galois theory for the Selmer group of an abelian variety, Compos. Math. 136 (2003), 255–297.CrossRefGoogle Scholar
Greenberg, R.. On the structure of Selmer groups. In Elliptic Curves, Modular Forms and Iwasawa Theory (in honour of John H. Coates’ 70th birthday; eds. Loeffler, D., Zerbes, S.L.), Springer Proc. in Math. and Stat. 188 (Springer, 2016), pp. 225-252.CrossRefGoogle Scholar
Hachimori, Y., Matsuno, K.. An analogue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebraic Geometry 8 (1999), 581–601.Google Scholar
Hachimori, Y., Matsuno, K.. On finite $\Lambda$ -submodules of Selmer groups of elliptic curves, Proc. Amer. Math. Soc. 128 (9) (2000), 2539–2541.CrossRefGoogle Scholar
Hachimori, Y., Ochiai, T.. Notes on non-commutative Iwasawa theory, Asian J. Math. 14 (1) (2010), 11–18.CrossRefGoogle Scholar
Hachimori, Y., Venjakob, O.. Completely faithful Selmer groups over Kummer extensions, Doc. Math. 2003, Extra Vol., 443–478.Google Scholar
Harris, M.. Correction to: “p-adic representations arising from descent on abelian varieties”, Compos. Math. 121 (1) (2000), 105–108.CrossRefGoogle Scholar
Howson, S.. Euler characteristics as invariants of Iwasawa modules, Proc. Lond. Math. Soc. 85 (3), (2002) 634–658.CrossRefGoogle Scholar
Iwasawa, K.. On the $\mu$ -invariants of ${{\mathbb{Z}}}_l$ -extensions. In Number Theory, Algebraic Geometry and Commutative Algebra (in honor of Yasuo Akizuki), (Kinokuniya, Tokyo, 1973) pp. 1-11.Google Scholar
Kida, M.. Ramification in the division fields of an elliptic curve, Abh. Math. Sem. Univ. Hamburg 73 (2003), 195–207.CrossRefGoogle Scholar
Kida, M.. Variation of the reduction type of elliptic curves under small base change with wild ramification, Cent. Eur. J. Math. 1 (4) (2003), 510–560.Google Scholar
Kim, C.-H., Pollack, R. and Weston, T.. On the freeness of anticyclotomic Selmer groups of modular forms, Int. J. Number Theory, 13 (6) (2017), 1443–1455.CrossRefGoogle Scholar
Kato, K.. p-adic Hodge theory and values of zeta functions of modular forms. In Cohomologies p-adique et Applications Arithmetiques III. Astérisque 295 (2004), 117–290.Google Scholar
Kleine, S.. Local behavior of Iwasawa’s invariants, Int. J. Number Theory, 13 (4) (2017), 1013-1036.CrossRefGoogle Scholar
Kleine, S.. Bounding the Iwasawa invariants of Selmer groups, Canad. J. Math. 73 (5) (2021), 1390–1422.Google Scholar
Kleine, S.. On the Iwasawa invariants of non-cotorsion Selmer groups, Asian J. Math., 26 (3) (2022), 373–406.CrossRefGoogle Scholar
Kleine, S. and Matar, A.. Boundedness of Iwasawa Invariants of fine Selmer Groups and Selmer groups, Results Math., 78 (4) (2023), paper no. 148.CrossRefGoogle Scholar
Kidwell, K.. On the structure of Selmer groups of p-ordinary modular forms over ${{\mathbb{Z}_p}}$ -extensions. J. Number Theory, 187 (2018), 296–331.CrossRefGoogle Scholar
Kundu, D., Lei, A., Ray, A.. Arithmetic statistics and noncommutative Iwasawa theory, Doc. Math. 27 (2022), 89–149.Google Scholar
Kundu, D. and Ray, A.. Statistics for Iwasawa invariants of elliptic curves, Trans. Amer. Math. Soc. 374 (2021), 7945–7965.CrossRefGoogle Scholar
Lim, M.F.. A remark on the $\mathfrak{M}_H(G)$ -conjecture and Akashi series, Int. J. Number Theory 11 (1) (2015), 269–297.CrossRefGoogle Scholar
Lim, M.F.. Notes on the fine Selmer groups, Asian J. Math. 21 (2) (2017), 337-362.CrossRefGoogle Scholar
Lim, M. F.. Some remarks on Kida’s formula when $\mu \neq 0$ , The Ramanujan Journal 55 (3) (2021), 1127-1144.CrossRefGoogle Scholar
Matsumura, H.. Commutative Ring Theory (Cambridge University Press, 1989).Google Scholar
Mazur, B.. Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266.CrossRefGoogle Scholar
Mazur, B.. Modular Curves and Arithmetic, Proc. Internat. Congress Math., Vols. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 185–211.Google Scholar
Minardi, J.. Iwasawa Modules for $\mathbb{Z}^d_p$ -Extensions of Algebraic Number Fields, PhD. thesis. University of Washington (1986).Google Scholar
Monsky, P.. Some invariants of $\mathbb{Z}_p^d$ -extensions, Math. Ann. 255 229–233, 1981.CrossRefGoogle Scholar
Neukirch, J., Schmidt, A., Wingberg, K.. Cohomology of Number Fields, 2nd ed. Grundlehren Math. Wiss. 323, (Springer, 2008), xvi+825.CrossRefGoogle Scholar
Ono, K., Papanikolas, M.. Quadratic twists of modular forms and elliptic curves, Number Theory for the Millennium III (Urbana, IL, 2000), A. K. Peters, 2002, pp. 73-85.CrossRefGoogle Scholar
Pollack, R. and Weston, T.. On anticyclotomic $\mu$ -invariants of modular forms, Compositio. Math. 147 (5) (2011), 1353–1381.CrossRefGoogle Scholar
Ribes, L. and Zalesskii, P.. Profinite Groups. Ergeb. der Math. 40 (Springer-Verlag, 2000).Google Scholar
Rohrlich, D.. On L-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409–423.CrossRefGoogle Scholar
Rubin, K.. Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer. Lecture Notes in Math. 1716 (Springer, New York 1999), pp.167-234.Google Scholar
SageMath, the Sage Mathematics Software System (Version 8.9). The Sage Developers (2019), http://www.sagemath.org.Google Scholar
Silverman, J.. The Arithmetic of Elliptic Curves. Graduate Texts in Math. 106 (Springer-Verlag, 2009).CrossRefGoogle Scholar
Vatsal, V.. Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), 219–261.CrossRefGoogle Scholar
Venjakob, O.. Characteristic elements in non-commutative Iwasawa theory, J. Reine Angew. Math. 583 (2005), 193–236.CrossRefGoogle Scholar
Zarhin, Y.. Endomorphisms and torsion of abelian varieties, Duke Math. J. 54(1) (1987), 131–145.CrossRefGoogle Scholar
Zerbes, S.. Generalised Euler characteristics of Selmer groups, Proc. London Math. Soc. 98(3) (2008), 775–796.CrossRefGoogle Scholar