论文标题
椭圆曲线的扭转群的生长在数字场上变化后的基础变化
Growth of torsion groups of elliptic curves upon base change from number fields
论文作者
论文摘要
给定一个数字字段$ f_0 $，其中不包含任何假想二次二次字段的希尔伯特课程，我们表明存在一个有效的可计算常数$ b：= b（f_0）\ in \ mathbb {z}^+$，以下是任何有限的$ l/f_0 $ copri $ [l： $ e _ {/f_0} $，$ l $ - 合理扭转子组$ e（l）[\ textrm {tors}] = e（f_0）[\ textrm {tors}] $。这概括了González-Jiménez和Najman的先前结果，超过$ F_0 = \ Mathbb {q} $。 为了展示这一点，我们还证明了mod-$ \ ell $ galois代表椭圆曲线的相对统一分裂性的结果。此外，我们表明，当我们允许$ f_0 $具有合理定义的CM时，主要结果的结论失败了，这是因为存在$ f_0 $ - 理性的质量基因，这些$ f_0 $ - 理性的同基因是任意较大的高级学位，满足某些一致性条件。
Given a number field $F_0$ that contains no Hilbert class field of any imaginary quadratic field, we show that under GRH there exists an effectively computable constant $B:=B(F_0)\in\mathbb{Z}^+$ for which the following holds: for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup $E(L)[\textrm{tors}]=E(F_0)[\textrm{tors}]$. This generalizes a previous result of González-Jiménez and Najman over $F_0=\mathbb{Q}$. Towards showing this, we also prove a result on relative uniform divisibility of the index of a mod-$\ell$ Galois representation of an elliptic curve over $F_0$. Additionally, we show that the main result's conclusion fails when we allow $F_0$ to have rationally defined CM, due to the existence of $F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain congruency conditions.
