Semistable abelian varieties with small division fields
| dc.creator | Brumer, Armand | |
| dc.creator | Kramer, Kenneth | |
| dc.date | 2002-07-22 | |
| dc.date.accessioned | 2026-07-07T04:49:58Z | |
| dc.date.available | 2026-07-07T04:49:58Z | |
| dc.description | Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(μ_\ell)$ be the cyclotomic field generated by the group of $\ell^{th}$-roots of unity. We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is "small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent. We show that if $\ell=2$ or 3 and $H$ is nilpotent then the reduction of $A$ at $p$ is totally toroidal, so its conductor is $p^{\dim A}$. The Jacobian of the modular curve $X_0(41)$ is a simple semistable abelian variety of dimension 3, with bad reduction only at $p=41$ and the Galois group of its 2-division field is a 2-group. For $\ell=2$, 3 or 5, there exist elliptic curves $E$ of prime conductor such that ${\bf Q}(E[\ell]) = {\bf Q}(μ_{2 \ell})$. We characterize the abelian varieties that are isogenous to products $E^d$. | |
| dc.identifier | https://arxiv.org/abs/math/0207309 | |
| dc.identifier | http://arxiv.org/abs/math/0207309 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64633 | |
| dc.subject | Number Theory | |
| dc.title | Semistable abelian varieties with small division fields | |
| dc.type | text |