论文标题
椭圆曲线的循环降低密度
Cyclic reduction densities for elliptic curves
论文作者
论文摘要
对于在数字字段$ k $上定义的椭圆曲线$ e $，$ e $ $ e $的启发式启发式密度是由包含 - 排名sum $ $ $δ_{e/k} $涉及$ m $ m $ m $ divivision fields $ k_m $ e $ $ e $ $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ g。在GRH的假设下，该密度可以证明是正确的。 对于不复杂乘法（CM）的$ e $，我们表明$δ_{e/k} $是明确的非负合理数字的产物，反映了$ e $的划分领域的有限纠缠和通用的无限Artin型产品。对于$ e $，通过二次订单$ {\ mathcal {o}} $承认$ k $的CM，我们表明$δ_{e/k} $接受类似的“分解”，其中Artin类型也取决于$ {\ Mathcal {O}} $。 对于$ e $，通过订单$ {\ Mathcal {o}} \ not \ subset k $承认$ \ bar k $的cm k $，以$ k = {\ bf q} $发生，$ k $以上的$ k $的纠缠是非限制的。在这种情况下，我们将$δ_{e/k} $编写为两种贡献的总和，这些贡献来自$ k $的primes，这些贡献是$ k $的，这些贡献是分裂的，并在$ {\ mathcal {o}} $中屈服。可以通过以前的方法来处理分裂贡献，惰性贡献的性质不同。 我们确定密度消失的方式，并提供不同类型的密度的数值示例。
For an elliptic curve $E$ defined over a number field $K$, the heuristic density of the set of primes of $K$ for which $E$ has cyclic reduction is given by an inclusion-exclusion sum $δ_{E/K}$ involving the degrees of the $m$-division fields $K_m$ of $E$ over $K$. This density can be proved to be correct under assumption of GRH. For $E$ without complex multiplication (CM), we show that $δ_{E/K}$ is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of $E$ and a universal infinite Artin-type product. For $E$ admitting CM over $K$ by a quadratic order ${\mathcal{O}}$, we show that $δ_{E/K}$ admits a similar `factorization' in which the Artin type product also depends on ${\mathcal{O}}$. For $E$ admitting CM over $\bar K$ by an order ${\mathcal{O}}\not\subset K$, which occurs for $K={\bf Q}$, the entanglement of division fields over $K$ is non-finite. In this case we write $δ_{E/K}$ as the sum of two contributions coming from the primes of $K$ that are split and inert in ${\mathcal{O}}$. The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities.
