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

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

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
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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
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 36
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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