A Heuristic approach to the Iwasawa theory of elliptic curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911677772988416 |
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| author | Müller, Katharina Ray, Anwesh |
| author_facet | Müller, Katharina Ray, Anwesh |
| contents | Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that $E$ has good ordinary reduction at $p$ and the Galois representation on $E[p]$ is irreducible. Then Greenberg's $μ=0$ conjecture predicts that the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ is cofinitely generated as a $\mathbb{Z}_p$-module. In this article we study this conjecture from a statistical perspective. We extend the heuristics of Poonen and Rains to obtain further evidence for Greenberg's conjecture. The key idea is that the vanishing of the $μ$-invariant can be detected by the intersection $M_1\cap M_2$ of two Iwasawa modules $M_1, M_2$ with additional properties in a given inner product space. The heuristic is based on showing that there is a probability measure on the space of pairs $(M_1, M_2)$ respect to which the event that $M_1\cap M_2$ is finite happens with probability $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_15056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Heuristic approach to the Iwasawa theory of elliptic curves Müller, Katharina Ray, Anwesh Number Theory Probability 11R23, 11G05, 11R45 Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that $E$ has good ordinary reduction at $p$ and the Galois representation on $E[p]$ is irreducible. Then Greenberg's $μ=0$ conjecture predicts that the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ is cofinitely generated as a $\mathbb{Z}_p$-module. In this article we study this conjecture from a statistical perspective. We extend the heuristics of Poonen and Rains to obtain further evidence for Greenberg's conjecture. The key idea is that the vanishing of the $μ$-invariant can be detected by the intersection $M_1\cap M_2$ of two Iwasawa modules $M_1, M_2$ with additional properties in a given inner product space. The heuristic is based on showing that there is a probability measure on the space of pairs $(M_1, M_2)$ respect to which the event that $M_1\cap M_2$ is finite happens with probability $1$. |
| title | A Heuristic approach to the Iwasawa theory of elliptic curves |
| topic | Number Theory Probability 11R23, 11G05, 11R45 |
| url | https://arxiv.org/abs/2409.15056 |