A bound on the $μ$-invariants of supersingular elliptic curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916586191847424 |
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| author | Gajek-Leonard, Rylan |
| author_facet | Gajek-Leonard, Rylan |
| contents | Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants $μ_p^\pm$ and $λ_p^\pm$ which encode arithmetic properties of $E$ along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $μ_p^\pm=0$. We provide support for this conjecture by proving that for any $\ell\geq 0$, we have $μ_p^\pm\leq 1$ for all but finitely many primes $p$ with $λ_p^\pm=\ell$. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $μ_p^\pm\leq 1$ holds on a density 1 set of good supersingular primes for $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A bound on the $μ$-invariants of supersingular elliptic curves Gajek-Leonard, Rylan Number Theory 11R23 Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants $μ_p^\pm$ and $λ_p^\pm$ which encode arithmetic properties of $E$ along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $μ_p^\pm=0$. We provide support for this conjecture by proving that for any $\ell\geq 0$, we have $μ_p^\pm\leq 1$ for all but finitely many primes $p$ with $λ_p^\pm=\ell$. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $μ_p^\pm\leq 1$ holds on a density 1 set of good supersingular primes for $E$. |
| title | A bound on the $μ$-invariants of supersingular elliptic curves |
| topic | Number Theory 11R23 |
| url | https://arxiv.org/abs/2409.18021 |