Certain families of K3 surfaces and their modularity

Abstract: We start with a double sextic family of K3 surfaces with four parameters with Picard number $16$. Then by geometric reduction (top-to-bottom) processes, we obtain three, two and one parameter families of K3 surfaces of Picard number $17, 18$ and $19$ respectively. All these families turn out to be of hypergeometric type in the sense that their Picard--Fuchs differential equations are given by hypergeometric or Heun functions. We will study the geometry of two parameter families in detail.

We will then prove, after suitable specializations of parameters, these K3 surfaces will have CM (complex multiplication), and will become modular in the sense that the Galois representations of dimensions $\leq 6$ associated to the transcendental lattices are all induced from $1$-dimensional representations. Thus, these K3 surfaces will be determined by modular forms of various weights. This is done starting with one-parameter family establishing the modularity at special fibers, and then applying arithmetic induction (bottom-to-top) processes to multi-parameter families. This is a joint work with A. Clingher, A. Malmendier, and N. Yui.

number theory

Audience: researchers in the topic

London number theory seminar

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