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arXiv · 2303.04373

Mazur's main conjecture at Eisenstein primes

Francesc Castella·Giada Grossi·Christopher Skinner
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Abstract

Let $E/\mathbb{Q}$ 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 $\mathbb{F}_p(\phi)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $\phi\vert_{G_p}\neq 1,\omega$, where $G_p$ is a decomposition group at $p$ and $\omega$ is the Teichm\"uller 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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Francesc Castella, Giada Grossi, Christopher Skinner. 2023-03-08. Mazur's main conjecture at Eisenstein primes. https://arxiv.org/abs/2303.04373

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