Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-31T05:00:22.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Congruent elliptic curves and the $\tau $-component in the Iwasawa main conjecture

Part of: Algebraic number theory: global fields Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  29 May 2025

RAIZA CORPUZ  and
DANIEL DELBOURGO
RAIZA CORPUZ
Affiliation:
School of Computing & Mathematical Sciences, University of Waikato, Gate 8, Hillcrest Road, Hamilton 3240, New Zealand. e-mail: rc202@students.waikato.ac.nz
DANIEL DELBOURGO
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand. e-mail: daniel.delbourgo@auckland.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p. Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F, induced from a Hecke character over a CM-extension $K/F$. Assuming $E_1$ and $E_2$ are ordinary at p, we compute the variation in the $\mu$- and $\lambda$-invariants for the $\tau$-part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$. Provided an Euler system exists, it will follow directly that IMC$(E_1,\tau)$ is true if and only if IMC$(E_2,\tau)$ is true.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23: Iwasawa theory

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 179 , Issue 2 , September 2025 , pp. 373 - 408
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bosma, W., Cannon, J. and Playoust, C.. The MAGMA Computer Algebra System, version 2.27-8, (University of Sydney, 1997).Google Scholar
Castella, F., Kim, C.-H. and Longo, M., Variation of anticyclotomic Iwasawa invariants in Hida families. Algebra Number Theory 11 (2017), 2339–2368.CrossRefGoogle Scholar
Choi, S. H. and Kim, B. D., Congruences of two-variable p-adic L-functions of congruent modular forms of different weights, The Ramanujan Journal 43 (2017), 163–195.10.1007/s11139-016-9819-8CrossRefGoogle Scholar
Corbett, A. and Saha, A., On the order of vanishing of newforms at cusps, Math. Res. Let. 25 (2018), 1771–1804.CrossRefGoogle Scholar
Corpuz, R.. Iwasawa theory for tensor products of Hilbert modular forms. PhD. thesis. University of Waikato (2025).Google Scholar
Delbourgo, D., Exceptional zeroes of p-adic L-functions over non-abelian field extensions, Glasg. Math. J. 58 (2016), 385–432.CrossRefGoogle Scholar
Delbourgo, D., Variation of the analytic $\lambda$ -invariant over a solvable extension, Proc. London Math. Soc. 120 (2020), 918–960.CrossRefGoogle Scholar
Delbourgo, D., Variation of the algebraic $\lambda$ -invariant over a solvable extension, Math. Proc. Camb. Phil. Soc, 170 (2021), 499–521.10.1017/S0305004119000495CrossRefGoogle Scholar
Delbourgo, D., On the Iwasawa $\mu$ -invariant and $\lambda$ -invariant associated to tensor products of newforms, Ann. Inst. Fourier 74 (2024), 451–502.CrossRefGoogle Scholar
Delbourgo, D. and Lei, A., Heegner cycles and congruences between anticyclotomic p-adic L-functions over CM-extensions, New York J. Math. 26 (2020), 496–525.Google Scholar
Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. 38 (2005), 505–551.CrossRefGoogle Scholar
Dimitrov, M., On Ihara’s lemma for Hilbert modular varieties, Compositio Math. 145 (2009), 1114–1146.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., Computations in non-commutative Iwasawa theory, Proc. London Math. Soc. 94 (2007), p 211–272.10.1112/plms/pdl014CrossRefGoogle Scholar
Emerton, M., Pollack, R. and Weston, T., Variation of Iwasawa invariants in Hida families, Invent. Math. 163 (2006), 523–580.10.1007/s00222-005-0467-7CrossRefGoogle Scholar
Gouvêa, F., Deforming Galois representations: controlling the conductor, J. Number Theory 34 (1990), 95–113.10.1016/0022-314X(90)90055-VCrossRefGoogle Scholar
Greenberg, R., Iwasawa theory for elliptic curves, In Arithmetic Theory of Elliptic Curves, Cetraro. 1997, Lecture Notes in Math. 1716 (1997), 51–144.10.1007/BFb0093453CrossRefGoogle Scholar
Greenberg, R. and Vatsal, V., On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), 17–63.10.1007/s002220000080CrossRefGoogle Scholar
Hida, H., On p-adic L-functions of GL(2) $\times$ GL(2) over totally real fields, Ann. Inst. Fourier 41 (1991), 311–391.10.5802/aif.1258CrossRefGoogle Scholar
Hida, H., On the critical values of L-functions of $\mathrm{GL}(2)$ and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ . D. Math. J. 74 (1994), 431–529.Google Scholar
Khare, C. and Wintenberger, J.-P., Serre’s modularity conjecture I, Invent. Math. 178 (2009), 485–504.10.1007/s00222-009-0205-7CrossRefGoogle Scholar
Lei, A., Loeffler, D. and Zerbes, S., Euler systems for Rankin-Selberg convolutions of modular forms, Ann. of Math. 180 (2014), 653–771.10.4007/annals.2014.180.2.6CrossRefGoogle Scholar
Livné, R.. On the conductors of mod l Galois representations coming from modular forms, J. of Number Theory 31 (1989), 133–141.10.1016/0022-314X(89)90015-2CrossRefGoogle Scholar
Kings, G., Loeffler, D. and Zerbes, S.. Rankin-Eisenstein classes and explicit reciprocity laws, Cambridge J. Math. 5 (2017), 1–122.10.4310/CJM.2017.v5.n1.a1CrossRefGoogle Scholar
Maletto, S.. An integral two variable $p$ -adic $L$ -function for the $\chi$ -twisted symmetric square representation associated to an ordinary modular form. PhD. thesis. University of British Columbia (2025).Google Scholar
Manning, J. and Shotton, J., Ihara’s Lemma for Shimura curves over totally real fields via patching, Math. Ann. 379 (2021), 187–234.10.1007/s00208-020-02048-8CrossRefGoogle Scholar
Panchishkin, A., Non-archimedean L-functions of Siegel and Hilbert modular forms. Lecture Notes in Math. 1471 (Springer-Verlag, 1991).10.1007/978-3-662-21541-8CrossRefGoogle Scholar
Ray, A., Sujatha, R. and Vatsal, V., Iwasawa invariants for symmetric square representations, Res. Math. Sci. 10 (27) (2023).CrossRefGoogle Scholar
Shemanske, T. and Walling, L., Twists of Hilbert modular forms, Trans. Amer. Math. Soc. 338 (1993), 375–403.10.1090/S0002-9947-1993-1102225-XCrossRefGoogle Scholar
Shimura, G., Special values of the zeta-functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), 637–679.10.1215/S0012-7094-78-04529-5CrossRefGoogle Scholar
Taguchi, Y., Induction formula for the Artin conductors of mod l Galois representations, Proc. Amer. Math. Soc. 130 (2002), 2865–2869.10.1090/S0002-9939-02-06524-3CrossRefGoogle Scholar