模椭圆曲线与双曲均匀化
Modular elliptic curves and hyperbolic uniformization
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中文总结 AI 辅助
研究将椭圆曲线双曲均匀化概念从有理数域扩展到奇数次完全实域，利用志村曲线，证明算术型双曲均匀化的存在意味着几何模性。
中文摘要 AI 辅助
在有理数域上椭圆曲线的模性被证明的几年前，Mazur将模性视为纯粹的复解析现象，定义了有理数域上具有算术型双曲均匀化的椭圆曲线概念。我们将这些想法扩展到奇数次完全实域上的椭圆曲线，特别证明了算术型双曲均匀化的存在意味着几何模性。
英文摘要
In an article published a few years before the modularity of elliptic curves over $\Q$ was proved, Mazur \cite{maz} looked at modularity as a purely complex analytic phenomenon, defining a notion of an elliptic curve over $\Q$ having a hyperbolic uniformisation of arithmetic type. Such an elliptic curve (of conductor $N$, say) is necessarily geometrically modular, i.e. a quotient of the jacobian of the modular curve $X_0(N)$, by a morphism defined over $\Q$. We extend these ideas to elliptic curves over totally real fields of odd degree, using Shimura curves for quaternion algebras split at all finite places and one real place. In particular, we prove that the existence of a hyperbolic uniformisation of arithmetic type would imply geometric modularity.
