Research on explicit constructions of motives associated to automorphic forms

(1)与某些刚性的Calabi-Yau三重相对应的Siegel准模形的构造。(2)与某些Siegel模簇相关的内窥镜升降器的构造及其L函数的计算。(3)在射影直线上具有连续Hodge数减三点且每个纤维具有位势自同构的Calabi-Yau族的构造。对于(3)，我们更具体地解释。从开曲线U：射影直线减三点上的一阶局部系统出发，利用卷积的方法，对每个自然数n，构造了秩为n+1，权为n的超几何层F_n/U，然后构造了一个Calabi-Yau族，当n为偶数时，它的中间上同调层使F_n达到代数循环.作为应用，我们证明了每根纤维的潜在自同构。

(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.