论文标题
在$ 12 $ - 椭圆形曲线
On $12$-congruences of elliptic curves
论文作者
论文摘要
我们构建了一对（几何非遗传性的）椭圆曲线对$ \ mathbb {q} $定义的无限族，其中$ 12 $ - torsion子组为Galois模块。这扩展了Chen和Fisher的先前工作，假定$ 12 $ torsion子组的基本同构尊敬的同构尊重Weil配对。我们的方法是为模块化对角线的表面计算明确的双向模型，以参数这对椭圆形曲线。 证明的关键要素是为$ 2 $，$ 3 $或$ 4 $ torsion的一对椭圆曲线的$ 2 $，$ 3 $或$ 4 $ torsion的条件构建简单（代数）条件。这些条件是根据一对椭圆曲线的$ j $ invariants给出的。
We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over $\mathbb{Q}$ with $12$-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of $12$-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the $2$, $3$, or $4$-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the $j$-invariants of the pair of elliptic curves.
