Volume 2013, Issue 677

The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G -torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p -group. We obtain similar formulas for the essential p -dimension of a broad class of groups, which includes all algebraic tori.

We study the structure of the invariant of K 3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K 3 lattice with respect to the involution is strictly bigger than 10, we prove that this automorphic form is expressed as the tensor product of an explicit Borcherds lift and Igusa's Siegel modular form.

In this paper we introduce the concept of weighted deficiency for abstract and pro- p groups and study groups of positive weighted deficiency which generalize Golod–Shafarevich groups. In order to study weighted deficiency, we introduce weighted versions of the notions of rank for groups and index for subgroups and establish weighted analogues of several classical results in combinatorial group theory, including the Schreier index formula. Two main applications of groups of positive weighted deficiency are given. First we construct infinite finitely generated residually finite p -torsion groups in which every finitely generated subgroup is either finite or of finite index—these groups can be thought of as residually finite analogues of Tarski monsters. Second we develop a new method for constructing just-infinite groups (abstract or pro- p ) with prescribed properties; in particular, we show that graded group algebras of just-infinite groups can have exponential growth. We also prove that every group of positive weighted deficiency has a hereditarily just-infinite quotient. This disproves a conjecture of Boston on the structure of quotients of certain Galois groups and solves Problem 15.18 from the Kourovka notebook.

We give a characterization of Drinfeld centers of fusion categories as non-degenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group.

Let G be a semisimple affine algebraic group of inner type over a field F . We write 𝔛 G for the class of all finite direct products of projective G -homogeneous F -varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in 𝔛 G . More precisely, it is known that the motive of any variety in 𝔛 G decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group PGL  A of automorphisms of a given central simple F -algebra A , for any variety in the class 𝔛 G (which includes the generalized Severi–Brauer varieties of the algebra A ) we determine its canonical dimension at any prime p . In particular, we find out which varieties in 𝔛 G are p - incompressible . If A is a division algebra of degree p n for some n  ≧ 0, then the list of p -incompressible varieties includes the generalized Severi–Brauer variety X ( p m ;   A ) of ideals of reduced dimension p m for m  = 0, 1, …,  n .

We consider algebraic Delone sets Λ in the Euclidean plane and address the problem of distinguishing convex subsets of Λ by X-rays in prescribed Λ-directions, i.e. directions parallel to lines through two different points of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u . It is shown that for any algebraic Delone set Λ there are four prescribed Λ-directions such that any two convex subsets of Λ can be distinguished by the corresponding X-rays. We further prove the existence of a natural number c Λ such that any two convex subsets of Λ can be distinguished by their X-rays in any set of c Λ prescribed Λ-directions. In particular, this extends a well-known result of Gardner and Gritzmann on the corresponding problem for planar lattices to nonperiodic cases that are relevant in quasicrystallography.

Let G be a semisimple linear algebraic group defined over the field ℂ, and let C be an irreducible smooth complex projective curve of genus at least three. We compute the Brauer group of the smooth locus of the moduli space of semistable principal G -bundles over C . We also compute the Brauer group of the moduli stack of principal G -bundles over C .