Rational points and cohomology of discriminant varieties

Let G be a finite unitary reflection group acting in a complex vector space . The discriminant varietyX_G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser of G in GL(V) acts on X_G; in this work we determine the action of on the cohomology of X_G. In the classical cases this amounts to computing the cohomology of X_G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.     

Minimal ‐crystals and isomorphism numbers of isosimple ‐crystals

In this paper we generalize minimal p‐divisible groups defined by Oort to minimal F‐crystals over algebraically closed fields of positive characteristic. We prove a structural theorem of minimal F‐crystals and give an explicit formula of the Frobenius endomorphism of the basic minimal F‐crystals that are the building blocks of the general minimal F‐crystals. We then use minimal F‐crystals to generalize minimal heights of p‐divisible groups and give an upper bound of the isomorphism numbers of F‐crystals, whose isogeny type are determined by simple F‐isocrystals, in terms of their ranks, Hodge slopes and Newton slopes.     

Results on undecidability of isomorphism of forms over polynomial rings

We show the undecidability of the question of isomorphism of forms over a polynomial ring $ R[t_1,\ldots,t_n] $, assuming a hypothesis about units in certain quaternion rings. Assuming this, it follows that isomorphisms of modules, and of affine algebraic varieties over R are undecidable.In memory of Gian-Carlo Rota     

The fibered isomorphism conjecture in L-theory

This is the first of three articles on the Fibered Isomorphism Conjecture of Farrell and Jones for L-theory. Here we prove the conjecture for several well-known classes of groups. In fact we consider a general class of groups satisfying certain conditions which includes the above classes of groups.     

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n  -Massey Conjecture for n≥3$n\ge 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.     

Periodic points, linearizing maps, and the dynamical Mordell-Lang problem

Under suitable hypotheses, we prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism . We also prove a version of the Mordell-Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.     

The Merkuriev-Suslin theorem for any semi-local ring

We introduce here a method which uses étale neighborhoods to extend results from smooth semi-local rings to arbitrary semi-local rings A by passing to the henselization of a smooth presentation of A. The technique is used to show that étale cohomology of A agrees with Galois cohomology, to show that the Merkuriev-Suslin theorem holds for A, and to describe torsion in K₂(A).     

On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds

We construct new examples of non-Kahlerian 1-convex threefolds X with exceptional set≅P₁ (resp. ≅F₂). Also the structure of Pic(X) will be studied. On the other hand, we shall investigate the quasi-projective structure of certain Kahlerian compactifiable 1-convex manifolds; particular attention will be given to 3-fold cases through concrete examples.     

The local index formula in semifinite von Neumann algebras II: The even case

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

Explicit Mumford isomorphism for hyperelliptic curves

Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.     

A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )$. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of ${\mathbb T}$ and show that no smooth genus 5 curve on such a surface can contain a $g^1_3$. We also describe explicitly the singular divisors in the linear system |2?|.     

Periods of orbits modulo primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α∈V(K) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin's primitive root conjecture.     

Application of covering techniques to families of curves

Much success in finding rational points on curves has been obtained by using Chabauty's Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty's Theorem does not directly apply to a curve , a recent modification has been to cover the rational points on by those on a covering collection of curves , obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty's Theorem applies to each . So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty's Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate-Shafarevich group killed by a bielliptic isogeny.     

Some model-theoretic correspondences between dimension groups and AF algebras

If A⊆B are structures for a first-order language S, A is said to be algebraically (existentially) closed in B just in case every positive existential (existential) S_A-sentence true in B_A is true in A_A. In 1976 Elliott showed that unital AF (‘approximately finite-dimensional’) algebras are classified up to isomorphism by corresponding dimension groups with order unit. This paper shows that one dimension group with order unit is algebraically (existentially) closed in another just in case the corresponding AF algebras, viewed as metric structures, fall in the same relation.     

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

The Farahat-Higman ring of wreath products and Hilbert schemes

We study the structure constants of the class algebra of the wreath products Γ_n associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on gives rise to the graded ring with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring . The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of . The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.