Mazur's main conjecture at Eisenstein primes

2023/03/08 by Castella, Francesc, Grossi, Giada, Skinner, Christopher
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.04373

Abstract

Let E/ℚ be an elliptic curve, let p>2 be a prime of good reduction for E, and assume that E admits a rational p-isogeny with kernel \mathbbF_p(ϕ). In this paper we prove the cyclotomic Iwasawa main conjecture for E, as formulated by Mazur in 1972, when ϕ\vert_Gp≠ 1,ω, where G_p is a decomposition group at p and ω is the Teichmüller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of E over an imaginary quadratic field K in which p splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes.

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