椭圆曲线CM-twist的Selmer群
Selmer groups of CM-twists of elliptic curves
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中文总结 AI 辅助
本文在全实数域上证明存在大量椭圆曲线的CM-twist使其Selmer群平凡，从而秩为零且Shafarevich-Tate群平凡，推广了James-Ono方法并融合Morrow与Takai的结果。
中文摘要 AI 辅助
设$F$为次数$g\leq5$的全实伽罗瓦数域，$\ell$为奇素数。设$E/F$为具有$F$-有理$\ell$阶点的椭圆曲线。在显式的算术与局部假设，以及模$\ell$的显式非消失条件下，我们证明存在$\gg_{F,E,\ell}\frac{X^{1/(2g)}}{\log X}$个全负平方类$d\in F^\times/(F^\times)^2$，满足$\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X$，且$\operatorname{Sel}_\ell(E^d,F)$平凡。此类twist具有秩零且Shafarevich--Tate群的$\ell$-部分平凡。该条件可通过有限计算验证，我们在一个例子中进行了计算。我们将James和Ono的方法从$\mathbb{Q}$提升到全实情形，结合Morrow的一个定理（该定理将此类twist的Selmer群与CM扩张$F(\sqrt{d})$的类群联系起来）以及Takai关于相对类数的不可整除性定理。Takai仅通过二次Hecke特征进行twist，而Morrow的条件是局部的；我们将twist论证推广到本原二次剩余类特征，以使两个结果相容。
英文摘要
Let $F$ be a totally real Galois number field of degree $g\leq5$ and let $\ell$ be an odd prime. Let $E/F$ be an elliptic curve with an $F$-rational point of order $\ell$. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo $\ell$, we prove that there are $\gg_{F,E,\ell}\frac{X^{1/(2g)}}{\log X}$ totally negative square classes $d\in F^\times/(F^\times)^2$ with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which $\operatorname{Sel}_\ell(E^d,F)$ is trivial. Such twists have rank zero and trivial $\ell$-part of the Shafarevich--Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from $\mathbb{Q}$ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension $F(\sqrt{d})$, with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.
