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Iwasawa invariants for elliptic curves over ℤp-extensions and Kida's formula

  • Debanjana Kundu ORCID logo EMAIL logo and Anwesh Ray ORCID logo
From the journal Forum Mathematicum


This paper aims at studying the Iwasawa λ-invariant of the p-primary Selmer group. We study the growth behavior of p-primary Selmer groups in p-power degree extensions over non-cyclotomic p-extensions of a number field. We prove a generalization of Kida’s formula in such a case. Unlike the cyclotomic p-extension, where all primes are finitely decomposed, in the p-extensions we consider primes may be infinitely decomposed. In the second part of this paper, we study the relationship of Iwasawa invariants with respect to congruences, obtaining refinements of the results of Greenberg, Vatsal and Kidwell. As an application, we provide an algorithm for constructing elliptic curves with large anticyclotomic λ-invariant. Our results are illustrated by explicit computation.

MSC 2010: 11R23

Communicated by Jan Bruinier


The first-named author acknowledges the support of the PIMS Postdoctoral Fellowship. We thank the referee for carefully reading our manuscript.


[1] C. Adelmann, The Decomposition of Primes in Torsion Point Fields, Lecture Notes in Math. 1761, Springer, Berlin, 2004. Search in Google Scholar

[2] S. Ahmed, C. Aribam and S. Shekhar, Root numbers and parity of local Iwasawa invariants, J. Number Theory 177 (2017), 285–306. 10.1016/j.jnt.2017.01.019Search in Google Scholar

[3] M. Bertolini, Selmer groups and Heegner points in anticyclotomic 𝐙p-extensions, Compos. Math. 99 (1995), no. 2, 153–182. Search in Google Scholar

[4] M. Bertolini and H. Darmon, Iwasawa’s main conjecture for elliptic curves over anticyclotomic p-extensions, Ann. of Math. (2) 162 (2005), no. 1, 1–64. 10.4007/annals.2005.162.1Search in Google Scholar

[5] A. Bhave, Analogue of Kida’s formula for certain strongly admissible extensions, J. Number Theory 122 (2007), no. 1, 100–120. 10.1016/j.jnt.2006.02.008Search in Google Scholar

[6] D. Brink, Prime decomposition in the anti-cyclotomic extension, Math. Comp. 76 (2007), no. 260, 2127–2138. 10.1090/S0025-5718-07-01964-3Search in Google Scholar

[7] R. Bröker, K. Lauter and A. V. Sutherland, Modular polynomials via isogeny volcanoes, Math. Comp. 81 (2012), no. 278, 1201–1231. 10.1090/S0025-5718-2011-02508-1Search in Google Scholar

[8] J. H. Bruinier, K. Ono and A. V. Sutherland, Class polynomials for nonholomorphic modular functions, J. Number Theory 161 (2016), 204–229. 10.1016/j.jnt.2015.07.002Search in Google Scholar

[9] J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), no. 1–3, 129–174. 10.1007/s002220050048Search in Google Scholar

[10] J. Coates and R. Sujatha, Galois Cohomology of Elliptic Curves, Narosa, New Delh, 2000. Search in Google Scholar

[11] C. Cornut, Mazur’s conjecture on higher Heegner points, Invent. Math. 148 (2002), no. 3, 495–523. 10.1007/s002220100199Search in Google Scholar

[12] D. A. Cox, Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication, John Wiley & Sons, New York, 2011. Search in Google Scholar

[13] T. Dokchitser and V. Dokchitser, A remark on Tate’s algorithm and Kodaira types, Acta Arith. 160 (2013), no. 1, 95–100. 10.4064/aa160-1-6Search in Google Scholar

[14] R. Greenberg, Iwasawa theory for elliptic curves, Arithmetic Theory of Elliptic Curves (Cetraro 1997), Lecture Notes in Math. 1716, Springer, Berlin (1999), 51–144. 10.1007/BFb0093453Search in Google Scholar

[15] R. Greenberg, Introduction to Iwasawa theory for elliptic curves, Arithmetic Algebraic Geometry (Park City 1999), IAS/Park City Math. Ser. 9, American Mathematical Society, Providence (2001), 407–464. 10.1090/pcms/009/06Search in Google Scholar

[16] R. Greenberg, Galois theory for the Selmer group of an abelian variety, Compos. Math. 136 (2003), no. 3, 255–297. 10.1023/A:1023251032273Search in Google Scholar

[17] R. Greenberg and V. Vatsal, On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), no. 1, 17–63. 10.1007/s002220000080Search in Google Scholar

[18] Y. Hachimori and K. Matsuno, An analogue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), no. 3, 581–601. Search in Google Scholar

[19] J. Hatley and A. Lei, Arithmetic properties of signed Selmer groups at non-ordinary primes, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 3, 1259–1294. 10.5802/aif.3270Search in Google Scholar

[20] S. Howson, Iwasawa theory of Elliptic Curves for p-adic Lie extensions, PhD thesis, University of Cambridge, 1998. Search in Google Scholar

[21] K. Iwasawa, Riemann–Hurwitz formula and p-adic Galois representations for number fields, Tohoku Math. J. (2) 33 (1981), no. 2, 263–288. 10.2748/tmj/1178229453Search in Google Scholar

[22] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques. III, Astérisque 295, Société Mathématique de France, Paris (2004), 117–290. Search in Google Scholar

[23] Y. Kida, l-extensions of CM-fields and cyclotomic invariants, J. Number Theory 12 (1980), no. 4, 519–528. 10.1016/0022-314X(80)90042-6Search in Google Scholar

[24] K. Kidwell, On the structure of Selmer groups of p-ordinary modular forms over 𝐙p-extensions, J. Number Theory 187 (2018), 296–331. 10.1016/j.jnt.2017.11.002Search in Google Scholar

[25] B. D. Kim, The Iwasawa invariants of the plus/minus Selmer groups, Asian J. Math. 13 (2009), no. 2, 181–190. 10.4310/AJM.2009.v13.n2.a2Search in Google Scholar

[26] B. D. Kim, The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc. 95 (2013), no. 2, 189–200. 10.1017/S1446788713000165Search in Google Scholar

[27] C.-H. Kim, R. Pollack and T. Weston, On the freeness of anticyclotomic Selmer groups of modular forms, Int. J. Number Theory 13 (2017), no. 6, 1443–1455. 10.1142/S1793042117500804Search in Google Scholar

[28] D. Kundu, An analogue of Kida’s formula for fine Selmer groups of elliptic curves, J. Number Theory 222 (2021), 249–261. 10.1016/j.jnt.2020.12.009Search in Google Scholar

[29] D. Kundu and A. Ray, Anticyclotomic μ-invariants of residually reducible Galois representations, J. Number Theory 234 (2022), 476–498. 10.1016/j.jnt.2021.06.030Search in Google Scholar

[30] K. Matsuno, An analogue of Kida’s formula for the p-adic L-functions of modular elliptic curves, J. Number Theory 84 (2000), no. 1, 80–92. 10.1006/jnth.2000.2510Search in Google Scholar

[31] K. Matsuno, Construction of elliptic curves with large Iwasawa λ-invariants and large Tate–Shafarevich groups, Manuscripta Math. 122 (2007), no. 3, 289–304. 10.1007/s00229-006-0068-9Search in Google Scholar

[32] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. 10.1007/BF01389815Search in Google Scholar

[33] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, Grundlehren Math. Wiss. 323, Springer, Berlin, 2000. Search in Google Scholar

[34] B. Perrin-Riou, Fonctions Lp-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France 115 (1987), no. 4, 399–456. 10.24033/bsmf.2085Search in Google Scholar

[35] R. Pollack and T. Weston, Kida’s formula and congruences, Doc. Math. 2006 (2006), 615–630. Search in Google Scholar

[36] R. Pollack and T. Weston, On anticyclotomic μ-invariants of modular forms, Compos. Math. 147 (2011), no. 5, 1353–1381. 10.1112/S0010437X11005318Search in Google Scholar

[37] A. Ray and R. Sujatha, Euler characteristics and their congruences for multi-signed Selmer groups, Canad. J. Math. 1 (2020), 1–25. 10.4153/S0008414X21000699Search in Google Scholar

[38] K. Rubin and A. Silverberg, Families of elliptic curves with constant mod p representations, Elliptic Curves, Modular Forms, & Fermat’s Last Theorem (Hong Kong 1993), International Press, Boston (1995), 148–161. Search in Google Scholar

[39] J.-P. Serre, Abelian l-adic Representations and Elliptic Curves Res. Notes Math. 7, A K Peters, Wellesley, 1998. 10.1201/9781439863862Search in Google Scholar

[40] J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Grad. Texts in Math. 106, Springer, Dordrecht, 2009. 10.1007/978-0-387-09494-6Search in Google Scholar

[41] V. Vatsal, Uniform distribution of Heegner points, Invent. Math. 148 (2002), no. 1, 1–46. 10.1007/s002220100183Search in Google Scholar

[42] V. Vatsal, Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), no. 2, 219–261. 10.1215/S0012-7094-03-11622-1Search in Google Scholar

[43] L. C. Washington, Introduction to Cyclotomic Fields, Grad. Texts in Math. 83, Springer, New York, 1997. 10.1007/978-1-4612-1934-7Search in Google Scholar

[44] Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020, Search in Google Scholar

Received: 2021-08-05
Revised: 2022-02-24
Published Online: 2022-04-20
Published in Print: 2022-07-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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