Sho Tanimoto

Brauer Groups on K3 Surfaces and Arithmetic Applications

For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice T S of S; we classify these lattices up to isomorphism using Nikulin’s discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two, accounted for by transcendental Brauer-Manin obstructions.

On the geometry of thin exceptional sets in Manin’s conjecture

Manins Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manins Conjecture is a thin set.

Effective bounds for Brauer groups of Kummer surfaces over number fields

We study effective bounds for Brauer groups of Kummer surfaces associated to the Jacobians of curves of genus

The distribution of S-integral points on SL2-orbit closures of binary forms

2

Balanced line bundles on Fano varieties

defined over number fields.

Kodaira dimension of moduli of special cubic fourfolds

We study the distribution of

Balanced Line Bundles and Equivariant Compactifications of Homogeneous Spaces

S

Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups

-integral points on

Geometric Manin's Conjecture and rational curves

\mathrm{SL}_2

Geometric consistency of Manin's Conjecture

-orbit closures of binary forms and prove an asymptotic formula for the number of