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arXiv · 2307.12053

A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes

Antonio Lei
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Abstract

Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field $K$ such that $p$ is inert in $K$ and that $E$ has good reduction at $p$. Let $K_\infty$ be the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Agboola--Howard defined Kobayashi-type signed Selmer groups of $E$ over $K_\infty$ and showed that exactly one of them is cotorsion over the corresponding Iwasawa algebra. In this short note, we discuss a link between the characteristic ideals of the cotorsion signed Selmer group and the fine Selmer group building on a recent breakthrough of Burungale--Kobayashi--Ota on the structure of local points.

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Antonio Lei. 2023-07-22. A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes. https://doi.org/10.1093/qmath%2Fhaad028

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