论文标题
在某些长几阶的超级企业猜想上
On Some Hypergeometric Supercongruence Conjectures of Long
论文作者
论文摘要
2003年，罗德里格斯·维勒加斯（Rodriguez Villegas）猜想，作为某些刚性calabi-yau三倍的家族的时期，产生的高几幅功能之间的14个超级企业和重量4模块化形式的傅立叶系数。 Long，Tu，Yui和Zudilin在2019年给出了这些超级美食的统一证明。使用DWORD的P-ADIC技术，它们将原始的超级企业减少到仅涉及超几何序列的相关一致性。我们概括了他们的技术，以考虑最近由Long猜想的六个六个超级企业。特别是，我们证明了这六个案例中的每一个，都证明了长，tu，yui和Zudilin减少的一致性。我们还猜想了DWork的工作的概括，该工作已在计算上观察到，这将与与我们超几何数据相关的GALOIS表示的模块化证明，并提供了Long的猜想的全部证明。
In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version of Long, Tu, Yui, and Zudilin's reduced congruences for each of these six cases. We also conjecture a generalization of Dwork's work which has been observed computationally and which would, together with a proof of modularity for Galois representations associated to our hypergeometric data, yield a full proof of Long's conjectures.
