The Iwasawa $μ$-invariants of Elliptic Curves over $\mathbb{Q}$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913565971054592 |
|---|---|
| author | Chakravarthy, Adithya |
| author_facet | Chakravarthy, Adithya |
| contents | In this paper, we discuss a longstanding conjecture of Greenberg in the Iwasawa theory of elliptic curves. Greenberg's conjecture states that if $E/\mathbb{Q}$ is an elliptic curve with good ordinary reduction at $p$, and $E[p]$ is irreducible as a Galois module, then the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ has $μ$-invariant zero. We prove that if $E$ is an elliptic curve over $\mathbb{Q}$, then we have $μ\leq 1$ for all but finitely many primes $p$ of good ordinary reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07826 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Iwasawa $μ$-invariants of Elliptic Curves over $\mathbb{Q}$ Chakravarthy, Adithya Number Theory In this paper, we discuss a longstanding conjecture of Greenberg in the Iwasawa theory of elliptic curves. Greenberg's conjecture states that if $E/\mathbb{Q}$ is an elliptic curve with good ordinary reduction at $p$, and $E[p]$ is irreducible as a Galois module, then the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ has $μ$-invariant zero. We prove that if $E$ is an elliptic curve over $\mathbb{Q}$, then we have $μ\leq 1$ for all but finitely many primes $p$ of good ordinary reduction. |
| title | The Iwasawa $μ$-invariants of Elliptic Curves over $\mathbb{Q}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.07826 |