Near coincidences and nilpotent division fields
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918406791364608 |
|---|---|
| author | Daniels, Harris Rouse, Jeremy |
| author_facet | Daniels, Harris Rouse, Jeremy |
| contents | Let $E/\mathbb{Q}$ be an elliptic curve. We say that $E$ has a near coincidence of level $(n,m)$ if $m \mid n$ and $\mathbb{Q}(E[n]) = \mathbb{Q}(E[m],ζ_{n})$. We classify near coincidences of prime power level and use this result to give a classification of values of $n$ for which ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is a nilpotent group. Along the way we prove a Gauss-Wantzel analog for the elliptic curve $E\colon y^2 = x^3-x$, showing that $\mathbb{Q}(E[n])/\mathbb{Q}$ is constructible if and only if $φ(n)$ is a power of 2. Assuming that there are no non-CM rational points on the modular curves $X_{ns}^{+}(p)$ for primes $p > 11$, we show that ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ nilpotent implies that $n$ is a power of $2$ or $n \in \{ 3, 5, 6, 7, 15, 21 \}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Near coincidences and nilpotent division fields Daniels, Harris Rouse, Jeremy Number Theory Primary: 11G05, Secondary: 11R32, 14H52 Let $E/\mathbb{Q}$ be an elliptic curve. We say that $E$ has a near coincidence of level $(n,m)$ if $m \mid n$ and $\mathbb{Q}(E[n]) = \mathbb{Q}(E[m],ζ_{n})$. We classify near coincidences of prime power level and use this result to give a classification of values of $n$ for which ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is a nilpotent group. Along the way we prove a Gauss-Wantzel analog for the elliptic curve $E\colon y^2 = x^3-x$, showing that $\mathbb{Q}(E[n])/\mathbb{Q}$ is constructible if and only if $φ(n)$ is a power of 2. Assuming that there are no non-CM rational points on the modular curves $X_{ns}^{+}(p)$ for primes $p > 11$, we show that ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ nilpotent implies that $n$ is a power of $2$ or $n \in \{ 3, 5, 6, 7, 15, 21 \}$. |
| title | Near coincidences and nilpotent division fields |
| topic | Number Theory Primary: 11G05, Secondary: 11R32, 14H52 |
| url | https://arxiv.org/abs/2409.00881 |