Computational aspects of motivic Hadamard products

点击翻译按钮获取中文摘要

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.