Let A$^′$be a complete characteristic (0,p) discrete valuation ring with absolute ramification degree e and a perfect residue field. We are interested in studying the category${\mathcal F}$${\mathcal F}$$_A$$_′$of finite flat commutative group schemes over A$^′$withp-power order. When e= 1, Fontaine formulated the purely ’linear algebra‘ notion of a finite Honda system over A$^′$and constructed an anti-equivalence of categories between${\mathcal F}$${\mathcal F}$$_A$$_′$and the category of finite Honda systems over A$^′$ when p< 2. We generalize this theory to the case e≤ − 1.

Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the ’Cohen and Lenstra Heuristics‘ give a probability that p[nmid]h(D). If p>3 is prime, then subject to a mild condition, we show that $\# \{0<D<X|p\nmid h(D)\}\gg_p \frac{\sqrt{X}}{\log X}.$ This condition holds for each 3<p<5000.

Let X be the Fermat curve of degree q+1 over the field k of q$^2$ elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.

The purpose of this paper is to show how generalizations of generic vanishing theorems to a${\mirrored Q}$-divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).

In this paper, constructions of Jordan algebras over commutative rings are given which place, within a general set-up, the classical Tits constructions of exceptional central simple Jordan algebras over fields. These are used to exhibit nontrivial Jordan algebra bundles over the affine plane with a given exceptional Jordan division algebra over k as the fibre. The associated principal F$_4$ bundles are shown to admit no reduction of the structure group to any proper connected reductive subgroup.

We define a generalization of the Euler characteristic of a perfect complex of modules for the group ring of a finite group. This is combined with work of Lichtenbaum and Saito to define an equivariant Euler characteristic for G on regular projective surfaces over Z having a free action of a finite group. In positive characteristic we relate the Euler characteristic of G to the leading terms of the expansions of L-functions at s=1.

We analyzes a notion of strong semistability of principal G-bundles by including reduction to nonreduced parabolic subgroup schemes. It turns out that strong semistability is equivalent to the Frobenius semistability of Ramanan and Rananathan. We also give a bound for nonstrongly semsitability of a semistable GL(n)-bundle improving a previous result of Shepherd-Barron.

Let f be a complex polynomial mapping. We relate the behaviour of f ’at infinity‘ to the characteristic cycle associated to the projective closures of fibres of f. We obtain a condition on the characteristic cycle which is equivalent to a condition on the asymptotic behaviour of some of the minors of the Jacobian matrix of f. This condition generalizes the condition in the hypersurface case known as Malgrange‘s condition. The relation between this condition and the behavior of the characteristic cycle is a partial generalization of Parusinski‘s result in the hypersurface case. We show that the new condition implies the C$^∞$-triviality of f.

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini–Study metric from${\mirrored P}$$_2$(${\mirrored C}$) and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.

On P$^3$, we show that mathematical instantons in characteristic two are unobstructed. We produce upper bounds for the dimension of the moduli space of stable rank two bundles on P$^3$ in characteristic two. In cases where there is a phenomenon of good reduction modulo two, these give similar results in characteristic zero. We also give an example of a nonreduced component of the moduli space in characteristic two.

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

In this paper we provide a complete characterization for when the Rees algebra and the associated graded ring of a perfect Gorenstein ideal of grade three are Cohen–Macaulay. We also treat the case of second analytic deviation one ideals satisfying some mild assumptions. In another set of results we give criteria for an ideal to be of linear type. Finally, we describe the equations defining the Rees algebras of certain Northcott ideals.

We construct in all characteristics p7lt;2 a complete surface in the moduli space of smooth genus 6 curves. The surface is contained in the locus of curves with automorphisms.

We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.