Mass formulas and the basic locus of unitary Shimura varieties

In this article we compute the mass associated to any unimodular lattice in a Hermitian space over an arbitrary CM field under a condition at 2. We study the geometry and arithmetic of the basic locus of the GU(r,s)-Shimura variety associated to an imaginary quadratic field modulo a good prime p>2. We give explicit formulas for the numbers of irreducible and connected components of the basic locus, and of points of the zero-dimensional Ekedahl-Oort (EO) stratum, as well as of the irreducible components of basic EO strata when the signature is either (1, n-1) or (2,2).Comment: 47 pages, comments are welcom

On the supersingular locus of Shimura varieties for quaternionic unitary groups

We study a Shimura variety attached to a unitary similitude group of a skew-Hermitian form over a totally indefinite quaternion algebra over a totally real number field. We give a necessary and sufficient condition for the existence of skew-Hermitian self-dual lattices. Under this condition we show that the superspecial locus in the fiber at $p$ of the associated Shimura variety is non-empty. We also give an explicit formula for the number of irreducible components of the supersingular locus when $p$ is odd and unramified in the quaternion algebra.Comment: 37 pages, comments welcom

偶数次元完全交叉の行列式と判別式

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 斎藤 毅, 東京大学教授 辻 雄, 東京大学教授 志甫 淳, 東京大学教授 寺杣 友秀, 東京大学准教授 三枝 洋一University of Tokyo(東京大学

The determinant of a double covering of the projective space and the discriminant of the branch locus : announcement (Algebraic Number Theory and Related Topics 2013)

"Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The determinant of the Galois action on the ell-adic cohomology of the middle degree of a proper smooth variety of even dimension defines a quadratic character of the absolute Galois group of the base field of the variety. In this announcement, we state that for a double covering of the projective space of even dimension, the character is computed via the square root of the discriminant of the defining polynomial of the covering. As a corollary, we deduce that the parity of a Galois permutation of the exceptional divisors on a del Pezzo surface can be computed by the discriminant