Tetrahedral elliptic curves and the local-global principle for isogenies
Tetrahedral elliptic curves and the local-global principle for isogenies
复制标题
四面体椭圆曲线和等基因的局部全局原理
DOI：
10.2140/ant.2014.8.1201
复制
发表时间：
2013
影响因子：
1.3
通讯作者：
J. Cremona
中科院分区：
文献类型：
--
作者：
Barinder S. Banwait;J. Cremona
研究了数域K上椭圆曲线l-同构存在性的局部-整体原理的失效问题。Sutherland证明了在$\mathbb{Q}$上只有一个失败，它发生在$l=7$和唯一的$j$-不变量，并且给出了当$K$不包含$l $th分圆域的二次子域时这种失败的分类。在本文中，我们提供了一个分类的失败的数域，这确实包含这个二次域，我们发现了一个新的“例外”来源，这种失败所产生的例外子群的$\mbox{PGL}_2（\mathbb{F}_l）$。通过构造两条模曲线，$X_{s}（5）$和$X_{S_4}（13）$的模型，我们发现了两类新的椭圆曲线族，它们的原理失效，并且我们证明了，对于二次域，不可能有其他例外失效。
We study the failure of a local-global principle for the existence of $l$-isogenies for elliptic curves over number fields $K$. Sutherland has shown that over $\mathbb{Q}$ there is just one failure, which occurs for $l=7$ and a unique $j$-invariant, and has given a classification of such failures when $K$ does not contain the quadratic subfield of the $l$'th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new `exceptional' source of such failures arising from the exceptional subgroups of $\mbox{PGL}_2(\mathbb{F}_l)$. By constructing models of two modular curves, $X_{\text{s}}(5)$ and $X_{S_4}(13)$, we find two new families of elliptic curves for which the principle fails, and we show that, for quadratic fields, there can be no other exceptional failures.