International Mathematics Research Notices

doi: 10.1093/imrn/rnt327pmid: N/A

doi: 10.1093/imrn/rnt303pmid: N/A

doi: 10.1093/imrn/rnt279pmid: N/A

Li, Han; Ng, Chi-Keung

doi: 10.1093/imrn/rnt097pmid: N/A

We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra L(). Moreover, a countable discrete group has property (T) if and only if for any action of on a von Neumann algebra N, every -invariant state on N is a weak--limit of a net of normal -invariant states.

Barroero, Fabrizio; Widmer, Martin

doi: 10.1093/imrn/rnt102pmid: N/A

Let be a lattice in , and let be a definable family in an O-minimal structure over . We give sharp estimates for the number of lattice points in the fibers . Along the way, we show that for any subspace of dimension j>0 the j-volume of the orthogonal projection of ZT to is, up to a constant depending only on the family Z, bounded by the maximal j-dimensional volume of the orthogonal projections to the j-dimensional coordinate subspaces.

Lee, Sangyop

doi: 10.1093/imrn/rnt087pmid: N/A

A twisted torus knot is a torus knot with a number of full twists on some adjacent strands. In this paper, we characterize twisted torus knots that are unknotted.

Fiorilli, Daniel

doi: 10.1093/imrn/rnt103pmid: N/A

We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over and the existence of highly biased elliptic curve prime number races. We show that conditionally on a Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of L(E,s), large analytic ranks translate into an extreme Chebyshev bias. Conversely, we show under a certain linear independence hypothesis on zeros of L(E,s) that if highly biased elliptic curve prime number races do exist, then the Riemann Hypothesis holds for infinitely many elliptic curve L-functions and there exist elliptic curves of arbitrarily large rank.

Lpez, Ricardo Garca

doi: 10.1093/imrn/rnt105pmid: N/A

We consider an analytic variant of the notion of Tate (or locally linearly compact) space and we show that, both in the complex and in the p-adic analytic setting, one can use it to define symbols which satisfy Weil-type reciprocity laws for curves.

Zywina, David

doi: 10.1093/imrn/rnt113pmid: N/A

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction Av of A modulo v splits up into isogeny. Assuming the MumfordTate conjecture for A and possibly increasing the field K, we will show that Av is isogenous to the mth power of an absolutely simple abelian variety for all places v of K away from a set of density 0, where m is an integer depending only on the endomorphism ring . This proves many cases, and supplies justification, of a conjecture of Murty and Patankar. Under the same assumptions, we will also describe the Galois extension of generated by the Weil numbers of Av for most v.

Fernando, Jos F.; Ueno, Carlos

doi: 10.1093/imrn/rnt112pmid: N/A

In this work we prove constructively that the complement of a convex polyhedron and the complement of its interior are regular images of . If K is moreover bounded, we can assure that and are also polynomial images of . The construction of such regular and polynomial maps is done by double induction on the number of facets (faces of maximal dimension) and the dimension of K; the careful placing (first and second trimming positions) of the involved convex polyhedra which appear in each inductive step has interest by its own and it is the crucial part of our technique.