Birational motives, II: Triangulated birational motives

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".Comment: Compared to the initial version: previous Subsection 4.2 has been upgraded to Section 5; previous Lemmas 5.2.5 and 5.2.6 have been corrected to Proposition 6.2.5 and Lemma 6.2.6; at the referee's request, previous Appendix B and the proof of previous Proposition C.1.1 (now A.4.1) have been removed (please consult the initial version for them

A few localisation theorems

Given a functor $T:C \to D$ carrying a class of morphisms $S\subset C$ into a class $S'\subset D$, we give sufficient conditions in order that $T$ induces an equivalence on the localised categories. These conditions are in the spirit of Quillen's theorem A. We give some applications in algebaic and birational geometry.Comment: File mistake in Version 2 To appear in Homology, Homotopy and Application

The Tate-Shafarevich group for elliptic curves with complex multiplication II

Let E be an elliptic curve over Q with complex multiplication. The aim of the present paper is to strengthen the theoretical and numerical results of \cite{CZS}. For each prime p, let t_{E/Q, p} denote the Z_p-corank of the p-primary subgroup of the Tate-Shafarevich group of E/Q. For each \epsilon 0, we prove that t_{E/Q, p} is bounded above by (1/2+\epsilon)p for all sufficiently large good ordinary primes p. We also do numerical calculations on one such E of rank 3, and 5 such E of rank 2, showing in all cases that t_{E/Q, p} = 0 for all good ordinary primes p < 30,000. In fact, we show that, with the possible exception of one good ordinary prime in this range for just one of the curves of rank 2, the p-primary subgroup of the Tate-Shafarevich group of the curve is zero (always supposing p is a good ordinary prime).Comment: 24 page

Symbolic-Connectionist Representational Model for Optimizing Decision Making Behavior in Intelligent Systems

Modeling higher order cognitive processes like human decision making come in three representational approaches namely symbolic, connectionist and symbolic-connectionist. Many connectionist neural network models are evolved over the decades for optimizing decision making behaviors and their agents are also in place. There had been attempts to implement symbolic structures within connectionist architectures with distributed representations. Our work was aimed at proposing an enhanced connectionist approach of optimizing the decisions within the framework of a symbolic cognitive model. The action selection module of this framework is forefront in evolving intelligent agents through a variety of soft computing models. As a continous effort, a Connectionist Cognitive Model (CCN) had been evolved by bringing a traditional symbolic cognitive process model proposed by LIDA as an inspiration to a feed forward neural network model for optimizing decion making behaviours in intelligent agents. Significanct progress was observed while comparing its performance with other varients

Grothendieck's theorem on non-abelian H^2 and local-global principles

A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization -- to the context of perfect fields of virtual cohomological dimension one -- takes the form of a local-global principle for the H^2-sets with respect to the orderings of the field. This principle asserts in particular that an element in H^2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k. Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over k is also given.Comment: 22 pages, AMS-TeX; accepted for publication by the Journal of the AM