Research on explicit constructions of motives associated to automorphic forms

(1)对应于某个刚性卡-丘三重的Siegel仿模形式的构造。(2)与某些Siegel模簇相关的内窥镜升降机的构造和该簇的L-函数的计算。(3)构造了一个在射影直线减三点上具有连续Hodge数且每根纤维都有势自同构的Calabi-Yau族.对于（3），我们给出了更具体的解释.从开曲线U上的一秩局部系：投影线减三点，利用卷积构造了秩为n+1，权为n的超几何层F_n/U，进而构造了一个Calabi-Yau族，当n为偶数时，该族的中上同调层实现了F_n的代数圈.作为一个应用，我们证明了每个纤维的潜在自同构。

(1) A construction of Siegel paramodular form corresponding to some rigid Calabi-Yau threefold.(2) A construction of endoscopic lifts associated to some Siegel modular variety and a computation of L-function of that variety.(3) A construction of a Calabi-Yau family having continuous Hodge number over the projective line minus three points and potential automorphy of each fiber.For (3), we explain more concretely. From local systems of rank one over the open curve U : projective line minus three points, by using convolution we construct hypergeometric sheaves F_n/U of rank n+1, weight n for each natural number n and then we construct a Calabi-Yau family whose middle cohomology sheave realizes F_n up to algebraic cycles when n is even. As an application, we prove potential automorphy of each fiber.