虚数乗法を持つK3曲面の整数論的性質とその応用について (Algebraic Number Theory and Related Topics 2017)

Abstract

This survey article is an outline of author's talk at the RIMS Workshop Algebraic Number Theory and Related Topics (2017). We study arithmetic properties of K3 surfaces with complex multiplication (CM) generalizing the results of Shimada for K3 surfaces with Picard number 20. Then, following Taelman's strategy and using Matsumoto's good reduction criterion for K3 surfaces with CM, we construct K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields. We also prove the Tate conjecture for self-products of K3 surfaces over finite fields by CM lifts and the Hodge conjecture for self-products of K3 surfaces with CM proved by Mukai and Buskin.