Computational aspects of motivic Hadamard products

两个幂级数f和g的通常Hadamard积f * g与复数乘法群上的卷积有关。如果f和g是motivic幂级数，因为它们描述了两个簇族的上同调群的周期的变化，那么f * g也是motivic的，通过在乘法群的平方上扭曲原始簇族。通过这种方式，人们可以得到阿达玛乘积动机的家族，这些动机在数学和物理学中的许多应用中都很重要。以下动机作为阿达玛积出现，说明了这类动机的重要性：广义超几何微分方程的动机，在量子力学的两体问题中发挥作用，以及出现在弦论镜像对称猜想中的卡-丘簇族的动机。后者的动机也发挥了重要作用，在最近的证明佐藤-泰特猜想。本建议的目的是开发和实施的算法计算的单值，霍奇型和伽罗瓦表示阿达玛产品的动机。由此产生的程序应适用于其他模块化问题（刚性）卡-丘品种和研究的多面体和费曼积分。

The usual Hadamard product f * g of two power series f and g is well known to be related to the convolution on the multiplicative group of the complex numbers. If f and g are motivic power series in the sense that they describe the variation of periods of cohomology groups of two family of varieties, then also f * g is motivic by twisting the original families over the square of the multiplicative group. In this way one obtains families of Hadamard product motives which are important for many applications in mathematics and physics. The following motives occur as Hadamard products and illustrate the importance of this class: Motives of generalized hypergeometric differential equations, playing a role in the two-body problem in quantum mechanics, and motives of families of Calabi-Yau-varieties occurring in the mirror symmetry conjecture of string theory. The latter motives also play an important role in the recent proof of the Sato-Tate conjecture. The aim of this proposal is the development and the implementation of algorithms for the computation of the monodromy, Hodge Type and Galois representations for Hadamard product motives. The resulting programs should be applied among others on modularity questions of (rigid) Calabi-Yau varieties and the study of polylogarithms and Feynman integrals.