Modularity of special motives of rank four associated with Calabi-Yau threefolds

Abstract

In this thesis, we study modularity properties of pure motives and mixed period matrices of rank four. The pure motives that we consider are associated with Calabi-Yau threefolds, while the mixed period matrices are associated with limit mixed Hodge structures of hypergeometric families of Calabi-Yau threefolds. By modularity properties, we mean two possible things. First, whether associated Galois representations are given by Galois representations of modular forms. And, second, whether associated period matrices are given by period matrices of modular forms. We do not only consider elliptic modular forms, but also Hilbert modular forms and Bianchi modular forms. <br /> After a short introduction about modularity properties of algebraic varieties, we review how one can associate motives with algebraic varieties in the second chapter. Here, we consider motives as purely linear algebraic structures which contain information about Galois representations and periods of the underlying variety. In the third chapter, we introduce elliptic modular forms and review that some of these have associated motives of rank two. As generalizations, we also discuss Hilbert modular forms and Bianchi modular forms. In the fourth chapter, we introduce Calabi-Yau threefolds and review how families of these can be studied using differential equations. <br /> We present new results in the last two chapters. This starts in the fifth chapter, where we consider four Calabi-Yau threefolds whose associated pure motives of rank four are (conjecturally) given by sums or products of motives associated with modular forms. While the results on the level of the periods are numerical, we can prove the modularity of the Galois representations in two cases. For instance, we give the first example of a Calabi-Yau threefold whose associated Galois representations are proven to be associated with Bianchi modular forms of weight 4 and weight 2. In the last chapter, we consider mixed period matrices associated with limit mixed Hodge structures of hypergeometric families of Calabi-Yau threefolds. It is known that there are fourteen such families and we study these using the method of "fibering out", which has recently been introduced by Vasily Golyshev. For twelve examples, we prove that the period matrices can be expressed completely in terms of integrals of modular forms. It has been expected that this is possible for a certain submatrix, but the result for the whole period matrix is surprising and leads to an interesting new class of periods of meromorphic modular forms. While our computations allow to give many examples of these modular forms and to prove their special properties, we do not have a more general understanding which is independent of the relation with families of Calabi-Yau threefolds. We conclude this thesis by giving experimental identities of a new type which relate mixed periods to central values of derivatives of L-functions.