On the supersingular reduction of K3 surfaces with complex multiplication

We study the good reduction modulo p of K3 surfaces with complex multiplication. If a K3 surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for K3 surfaces with Picard number 20. Our methods rely on the main theorem of complex multiplication for K3 surfaces by Rizov, an explicit description of the Breuil-Kisin modules associated with Lubin-Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.Comment: 29 pages, to appear in International Mathematics Research Notice

Nuclear dynamics in time-dependent picture

Using the time-dependent theory of quantum mechanics, we investigate nuclear electric dipole responses. The time evolution of a wave function is explicitly calculated in the coordinate-space representation. The particle continuum is treated with the absorbing boundary condition. Calculated time-dependent quantities are transformed into those of familiar energy representation. We apply the method to a three-body model for 11Li and to the mean-field model for 22O, then discuss properties of E1 response.Comment: 6 pages, 2 figures, Talk at the Sixth China Japan Joint Nuclear Physics Symposium, Shanghai, China, May 16-20, 200

Prismatic GG-display and descent theory

For a smooth affine group scheme $G$ over the ring of $p$-adic integers $\mathbb{Z}_p$ and a cocharacter $\mu$ of $G$, we study $G$-$\mu$-displays over the prismatic site of Bhatt-Scholze. In particular, we obtain several descent results for them. If $G=\mathrm{GL}_n$, then our $G$-$\mu$-displays can be thought of as Breuil-Kisin modules with some additional conditions. In fact, our results are formulated and proved for smooth affine group schemes over the ring of integers $\mathcal{O}_E$ of any finite extension $E$ of $\mathbb{Q}_p$ by using $\mathcal{O}_E$-prisms, which are $\mathcal{O}_E$-analogues of prisms.Comment: 60 pages, comments welcom

CM liftings of K3 surfaces over finite fields and their applications to the Tate conjecture

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over finite fields. We prove that every K3 surface of finite height over a finite field admits a characteristic 0 lifting whose generic fibre is a K3 surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a K3 surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a K3 surface of finite height and construct characteristic 0 liftings of the K3 surface preserving the action of tori in the algebraic group. We obtain these results for K3 surfaces over finite fields of any characteristics, including those of characteristic 2 or 3