Factorisation properties of group scheme actions

Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free commutative. From this we deduce, in certain special cases, results about the monoid of nonzero semi-invariants and the algebra of invariants. We use an infinitesimal method which allows us to work over an arbitrary base field.     

Nonsingular plane cubic curves over finite fields

We determine the number of projectively inequivalent nonsingular plane cubic curves over a finite field F_q with a fixed number of points defined over F_q. We count these curves by counting elliptic curves over F_q together with a rational point which is annihilated by 3, up to a certain equivalence relation.     

Monomial Base for Little q-Schur Algebra uk(2, r) at Even Roots of Unity

Let uk(2, r) be a little q-Schur algebra over k, where k is a field containing an l-th primitive root ε of 1 with l ≥ 4 even, the author constructs a certain monomial base for little q-Schur algebra uk(2, r).     

Base change for ≈SL2

Solvable base change for the metaplectic group ≈SL₂ over both local and global fields in studied. The method is to use Waldspurger's correspondence to relate base change for ≈SL₂ to base change for PGL₂. In the global case there are certain cuspidal representations which “disappear” under a quadratic base change. They then “reappear” under any subsequent quadratic base change. Such a phenomenon never occurs for PGL₂.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

Variations on a theme of rationality of cycles

We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2 ^r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.     

The ordinarity of an isotrivial elliptic fibration

In this paper, we study the ordinarity of an isotrivial elliptic surface defined over a field of positive characteristic. If an isotrivial elliptic fibration ?? : X ?? C is given, X is ordinary when the common fiber of ?? is ordinary and a certain finite cover of the base C is ordinary. By this result, we may obtain the ordinary reduction theorem for some kinds of isotrivial elliptic surfaces defined over a number field.     

Scalar Extensions of Triangulated Categories

Given a triangulated category ${{\mathcal T}}$ over a field K and a field extension L/K, we investigate how one can construct a triangulated category ${{\mathcal T}}_L$ over L. Our approach produces the derived category of the base change scheme X _L if ${{\mathcal T}}$ is the bounded derived category of a smooth projective variety over K and the field extension is finite and Galois. We also investigate how the dimension of a triangulated category behaves under scalar extensions.     

Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras

In this paper, we obtain a canonical central element ν_H for each semi-simple quasi-Hopf algebra H over any field k and prove that ν_H is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character takes only the values 0, 1 or −1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(ν_H) is the corresponding Frobenius-Schur indicator. We also prove an analog of a theorem of Larson-Radford for split semi-simple quasi-Hopf algebras over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(ν_H), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero.     

On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial is irreducible over the rationals for all n?1 and has Galois group A_n if n+1 is an odd square, and S_n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L_n^(α)(x) can be embedded in a double cover.     

The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

We define the algebraic fundamental group π ₁(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G? π₁(G) is exact.     

The strict topology in non-Archimedean vector-valued function spaces

Let F be a non-trivial complete non-Archimedean valued field. We study the strict topology β₀ on the space C_b(X,E) of all bounded continuous functions from a topological space X to a non-Archimedean F-locally convex space E over F. We also show that the dual of the space (C_b(X,E), β_o) is a certain space of E′-valued measures and we give a characterization of the equicontinuous subsets of this dual space.     

Primitive Algebras with Arbitrary Gelfand-Kirillov Dimension

We construct, for every real β ≥ 2, a primitive affine algebra with Gelfand-Kirillov dimension β. Unlike earlier constructions, there are no assumptions on the base field. In particular, this is the first construction over ℝ or L. Given a recursive sequence {v_n} of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in {v_n}. We bound the dimension of the resulting algebra in terms of the growth of {v_n}. In particular, if ⋎ν_n⋎ is less than doubly exponential, then the dimension is 2. This also answers affirmatively a conjecture of Salwa (1997, Comm. Algebra 25, 3965–3972).     

On the identities of the Grassmann algebras in characteristicp>0

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM ₂(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM ₂(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM ₂(K).     

Special values of generalized \mathbf{\lambda} functions at imaginary quadratic points

We study a modular function Λ _k,? that is one of generalized λ functions. We show that Λ _k,? and the modular invariant function j generate the modular function field with respect to the modular subgroup Γ ₁(N). Further, we prove that Λ _k,? is integral over Z[j]. From this result we obtain that a value of Λ _k,? at an imaginary quadratic point is an algebraic integer and generates a ray class field over a Hilbert class field.     

Intermediate subgroups in the chevalley groups of type B l , C l , F 4, and G 2 over the nonperfect fields of characteristic 2 and 3

We describe the intermediate subgroups in the Chevalley groups of type B _l , C _l , F ₄, and G ₂ over various fields of characteristic 2 and 3 in the case that the larger field is an algebraic extension of the smaller nonperfect field.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.