论文标题
椭圆曲线模块化参数化的分隔线
Divisors of Modular Parametrizations of Elliptic Curves
论文作者
论文摘要
模块化定理意味着，对于每个椭圆曲线$ e /\ mathbb {q} $，都有从模块化曲线$ x_0（n）$到$ e $的有理图，其中$ n $是$ e $的导体。这些地图可以用模块化功能对表示$ x（z）$和$ y（z）$，其中$ x（z）$和$ y（z）$满足$ e $的Weierstrass方程以及某些微分方程。使用这两种关系，可以使用递归算法来计算任何尖端的参数化的$ q $ - 扩展。 ％这些功能是$ \ mathbb {q}（j（z））$的代数，并且满足模块化多项式，其中每个系数函数都是$ j（z）$中的合理函数。 使用这些函数，我们确定参数化的除数和$ e $上的理性点的前图。当这些预映率对应于$ x_0（n）$上的CM点时，我们给出了足够的条件。我们还检查了这些功能对相关椭圆曲线产生的代数之间的连接，并描述了足够的条件以确定这些对象的$ q $ - 扩展中的一致性。
The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular functions $X(z)$ and $Y(z)$ where $X(z)$ and $Y(z)$ satisfy the Weierstrass equation for $E$ as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the $q$ - expansions of these parametrizations at any cusp. %These functions are algebraic over $\mathbb{Q}(j(z))$ and satisfy modular polynomials where each of the coefficient functions are rational functions in $j(z)$. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on $E$. We give a sufficient condition for when these preimages correspond to CM points on $X_0(N)$. We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the $q$-expansions of these objects.
