We classify non-symplectic automorphisms of 3-power order on algebraic $K3$ surfaces which act trivially on the Néron-Severi lattice, i.e., we describe their fixed locus. Moreover we give Weierstrass equations of $K3$ surfaces with a non-symplectic automorphism of 3-power order.

Feit and Thompson conjectured for distinct primes $p < q$ that $(q^{p}-1)/(q-1)$ never divides $(p^{q}-1)/(p-1)$. This paper is a record on partial solutions to this conjecture for small primes 3 and 5.

We investigate the problem of equipping a topology on cohomology groups (sets) in its relation with the problem of closedness of (relative) orbits for the action of algebraic groups on affine varieties defined over complete, especially $\mathfrak{p}$-adic fields and give some applications.

This note deals with a generalization of the famous Lyapunov inequality giving a necessary condition for the existence of solutions to a boundary value problem for an ordinary differential equation. The problem we consider is closely related to a well-known problem on an asymptotic behavior of positive solutions of a class of semilinear elliptic equations of nearly critical Sobolev growth.

We construct Néron models of intermediate Jacobians in degenerations over higher dimensional bases by means of log geometry.