AI 中文总结
研究改进显式超几何模性方法，开发适用于更广泛超几何数据的变体，确立特定超几何伽罗瓦表示的模性，构造相关模形式，还适合计算特殊L值，此前该三维模性已由他人用不同方法证明。
AI 中文摘要
我们改进了显式超几何模性方法（EHMM），并开发了一种适用于更广泛超几何数据类别的变体。作为应用，我们确立了来自长度为4的数据（不一定在有理数域上定义）的超几何伽罗瓦表示的模性。我们还用此方法明确构造了与罗德里格斯 - 维列加斯猜想为模性的超几何刚性卡拉比 - 丘三维相关的九个模形式。这些三维的模性最初由龙 - 涂 - 余 - 祖迪林基于法尔廷斯 - 塞尔方法用不同方法证明。此外，该方法的显式性质使其非常适合计算相关模形式的特殊L值。
英文摘要
We refine the Explicit Hypergeometric Modularity Method (EHMM) and develop a variant that applies to a broader class of hypergeometric data. As an application, we establish the modularity of hypergeometric Galois representations arising from length-$4$ data that are not necessarily defined over $\mathbb{Q}$. We also use this method to give explicit constructions of nine modular forms associated with hypergeometric rigid Calabi-Yau threefolds conjectured to be modular by Rodriguez-Villegas. The modularity of these threefolds was first proved by Long-Tu-Yui-Zudilin using a different approach based on the Faltings-Serre method. Moreover, the explicit nature of the method makes it well suited to the computation of special $L$-values of the associated modular forms.
