Positive elimination in valued fields

Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K ⁿ is again basic.     

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007     

Homogeneous spaces and degree 4 del Pezzo surfaces

It is known that, given a genus 2 curve , where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space for complete 2-descent on the Jacobian of , there is a V _δ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that . We shall prove that every degree 4 del Pezzo surface V, defined over K, arises in this way; furthermore, we shall show explicitly how, given V, to find and δ such that V =  V _δ , up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann’s example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over , whose Jacobians have nontrivial members of the Shafarevich-Tate group. This example will differ from previous examples in the literature by having only two -rational Weierstrass points. The author thanks EPSRC for support: grant number EP/F060661/1.     

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit of N is contained in the orbit closure and has codimension two. We show that the pointed variety is smoothly equivalent to a simple surface singularity of type A _n , or to the cone over a rational normal curve.     

Convergence of the Iterated Aluthge Transform Sequence for Diagonalizable Matrices II: λ-Aluthge Transform

Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U|T|. Then the λ-Aluthge transform is defined by

Notes on very ample vector bundles on 3-folds

Let be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of is provided when is nef but not big, and when a suitable positive multiple of defines a morphism X → B with connected fibers onto a smooth projective curve B, where K_X is the canonical bundle of X. As an application, the case where the genus of B is positive and has a global section whose zero locus is a smooth hyperelliptic curve of genus ≧ 2 is investigated, and our previous result is improved for threefolds. Received: 27 January 2005; revised: 26 March 2005     

On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary , and let be a covering map, where Y is a closed manifold. Let A _C be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace for N sufficiently large has a complete asymptotic expansion as . In particular, the heat trace has a complete asymptotic expansion as , and the -function has a meromorphic extension to .      

Fibred sites and stack cohomology

The usual notion of the site associated to a stack is expanded to a definition to a site fibred over a presheaf of categories A on a site . If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on . In particular, stack cohomology can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type.     

Intersecting subvarieties of $${G_m^n}$$ with algebraic subgroups

Let be an irreducible closed subvariety defined over . We bound the height of algebraic points on X that are in a certain sense close to the union of all algebraic subgroup of of dimension m < n/dim X. The bound we obtain is effective and will be expressed as a function of the height of X, the degree of X, and n. We then apply this bound to derive certain finiteness results if m is also strictly less than n − dim X.     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, a smooth geometrically connected curve defined over k with function field . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or >  2d + 1. Let p ≠ q be a prime number and a finite geometrically Galois and étale cover defined over k with function field . Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the -corank of the Selmer group Sel _p (A × _K K′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the -rank of the Lang–Néron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to , we give sufficient conditions for the Lang–Néron group of A to be uniformly bounded along the tower. This work was partially supported by CNPq research grant 305731/2006-8.     

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

The <Emphasis Type="Italic">L</Emphasis>-series of a cubic fourfold

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface S, defined over a number field K and satisfying , then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in terms of the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example.     

A transference principle for general groups and functional calculus on UMD spaces

Let –iA be the generator of a C ₀-group on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate in terms of the -Fourier multiplier norm of . If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev–de Laubenfels. If X is a UMD space, one obtains a bounded -calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded -calculus on sectors.     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

Local convergence of Newton’s method for subanalytic variational inclusions

This paper concerns variational inclusions of the form where f is a single locally Lipschitz subanalytic function and F is a set-valued map acting in Banach spaces. We prove the existence and the convergence of a sequence (x _k ) satisfying where lies to which is the Clarke Jacobian of f at the point x _k .      

Super-cocharacters,star-cocharacters and multiplicities bounded by one

Let A be an algebra over a field of characteristic zero with an additional structure of superalgebra or algebra with involution. The ordinary representation theory of the hyperoctahedral group is exploited in order to study the super-identities or the star-identities of A. One associates to A a sequence -characters and one of the main objective of the theory is to determine their decomposition into irreducibles. Here we classify the super-identities and the star-identities in case the corresponding multiplicities are bounded by one. This is strictly related to the varieties of algebras whose lattice of subvarieties is distributive. A. Giambruno was partially supported by MIUR of Italy. S. Mishchenko was partially supported by RFBR grant 07-01-00080.     

Clean rings of continuous functions

A ring is clean if every element is the sum of a unit and an idempotent. Let be a dense local subring of the reals which is not a field. We show that the ring of A-valued continuous functions on a zero-dimensional space X is clean if and only if X is a P-space, and examine some properties of the prime ideal spectrum of this ring. Received June 3, 2005; accepted in final form December 3, 2005.