Journal für die reine und angewandte Mathematik

Published by De Gruyter

Volume 2009 Issue 633

Issue of Journal für die reine und angewandte Mathematik

Contents
Journal Overview

June 16, 2009

The correspondence between a plane curve and its complement

Jérémy Blanc

Page range: 1-10

Abstract

Given two irreducible curves of the plane which have isomorphic complements, it is natural to ask whether there exists an automorphism of the plane that sends one curve on the other. This question has a positive answer for a large family of curves and H. Yoshihara conjectured that it is true in general. We exhibit counterexamples to this conjecture, over any ground field. In some of the cases, the curves are isomorphic and in others not; this provides counterexamples of two different kinds. Finally, we use our construction to find the existence of surprising non-linear automorphisms of affine surfaces.

June 16, 2009

Twisted conjugacy classes in nilpotent groups

Daciberg Gonçalves, Peter Wong

Page range: 11-27

Abstract

A group is said to have the R ∞ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R ∞ property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n ≧ 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the R ∞ property. The R ∞ property for virtually abelian and for -nilpotent groups are also discussed.

June 16, 2009

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Sebastian Casalaina-Martin, Radu Laza

Page range: 29-65

Abstract

A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.

June 16, 2009

Rank-one quadratic twists of an infinite family of elliptic curves

Dongho Byeon, Daeyeol Jeon, Chang Heon Kim

Page range: 67-76

Abstract

A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E /ℚ has (analytic) rank 1. This assertion has been confirmed by Vatsal [Math. Ann. 311: 791–794, 1998] and the first author [Acta Arith. 114: 391–396, 2004] for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E /ℚ using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.

June 16, 2009

An algebraic index theorem for Poisson manifolds

V. A. Dolgushev, V. N. Rubtsov

Page range: 77-113

Abstract

The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map.

June 16, 2009

Conformal symbols and the action of contact vector fields over the superline

Charles H. Conley

Page range: 115-163

Abstract

Let 𝒦 be the Lie superalgebra of contact vector fields on the supersymmetric line . We compute the action of 𝒦 on the modules of differential and pseudo-differential operators between spaces of tensor densities, in terms of their conformal symbols. As applications we deduce the geometric subsymbols, 1-cohomology, and various uniserial subquotients of these modules. We also outline the computation of the 𝒦-equivalences and symmetries of their subquotients.

June 16, 2009

A Galois correspondence for compact quantum group actions

Reiji Tomatsu

Page range: 165-182

Abstract

We establish a Galois correspondence for a minimal action of a compact quantum group 𝔾 on a von Neumann factor M . This extends the result of Izumi, Longo and Popa who treated the case of a Kac algebra. Namely, there exists a one-to-one correspondence between the lattice of left coideals of 𝔾 and that of intermediate subfactors of M 𝔾 ⊂ M .

June 16, 2009

Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Luis Dieulefait, Jorge Jiménez Urroz

Page range: 183-195

Abstract

We solve the diophantine equations x 4 + dy 2 = z p for d = 2 and d = 3 and any prime p > 349 and p > 131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat's Last Theorem, and by Ellenberg in the solution of the equation x 4 + y 2 = z p , and we use ℚ-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d .

June 16, 2009

Reduced zeta functions of Lie algebras

Anton Evseev

Page range: 197-211

Abstract

We define reduced zeta functions of Lie algebras, which can be derived, via the Euler characteristic, from motivic zeta functions counting subalgebras and ideals. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to analyse. We prove that reduced zeta functions are multiplicative under certain conditions and investigate which reduced zeta functions have functional equations.

June 16, 2009

Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

Goulnara Arzhantseva, Cornelia Druţu, Mark Sapir

Page range: 213-235

Abstract

We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. This answers a question of E. Guentner and G. Niblo. For a large class of Banach spaces ℰ (including all uniformly convex Banach spaces), the ℰ-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 2, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (moreover, of finite asymptotic dimension and exact) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.

About this journal

The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of "Crelle's Journal". In the 190 years of its existence, "Crelle's Journal" has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI's Journal Citation Report.