论文标题
具有非平凡扭转的有理椭圆曲线的最小模型
Minimal models of rational elliptic curves with non-trivial torsion
论文作者
论文摘要
在本文中，我们将所有椭圆曲线的最小判别因素明确分类为$ e/\ mathbb {q} $与非平凡的扭转子组。这是通过考虑各种椭圆曲线的参数化族来完成的，它们可以参数化所有椭圆曲线$ e/\ mathbb {q} $带有非平凡扭转点的属性。我们遵循这一点，通过可接受的变量更改，该变量为$ e $提供了全球最小模型。我们还提供了这些家庭参数的必要条件，以确定$ e $减少添加剂的素数。此外，我们使用这些参数化的家族为弗雷和屈肌 - 涂层而提供了新的结果证明，与数字$ k $相比的椭圆曲线，具有非客气$ k $ torsion点可以减少添加剂。
In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for $E$. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which $E$ has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterlé pertaining to the primes at which an elliptic curve over a number field $K$ with a non-trivial $K$-torsion point can have additive reduction.
