Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

For a property $\cal P$ of simplicial complexes, a simplicial complex $\Gamma$ is an obstruction to $\cal P$ if $\Gamma$ itself does not satisfy $\cal P$ but all of its proper restrictions satisfy $\cal P$. In this paper, we determine all obstructions to shellability of dimension $\le 2$, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions $\le 2$. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes

On the failure of pseudo-nullity of Iwasawa modules

We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p-adic Lie groups. The proof uses Kida's formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weakening of the usual assumption that mu is trivial. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.Comment: 26 page

The max-flow min-cut property of two-dimensional affine convex geometries

AbstractIn a matroid, (X,e) is a rooted circuit if X is a set not containing element e and X∪{e} is a circuit. We call X a broken circuit of e. A broken circuit clutter is the collection of broken circuits of a fixed element. Seymour [The matroids with the max-flow min-cut property, J. Combinatorial Theory B 23 (1977) 189–222] proved that a broken circuit clutter of a binary matroid has the max-flow min-cut property if and only if it does not contain a minor isomorphic to Q6. We shall present an analogue of this result in affine convex geometries. Precisely, we shall show that a broken circuit clutter of an element e in a convex geometry arising from two-dimensional point configuration has the max-flow min-cut property if and only if the configuration has no subset forming a ‘Pentagon’ configuration with center e.Firstly we introduce the notion of closed set systems. This leads to a common generalization of rooted circuits both of matroids and convex geometries (antimatroids). We further study some properties of affine convex geometries and their broken circuit clutters

On the Birch-Swinnerton-Dyer quotients modulo squares

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer rank and the analytic rank agree.Comment: 29 pages; minor changes; to appear in Annals of Mathematic

Saturated simplicial complexes

AbstractAmong shellable complexes a certain class has maximal modular homology, and these are the so-called saturated complexes. We extend the notion of saturation to arbitrary pure complexes and give a survey of their properties. It is shown that saturated complexes can be characterized via the p-rank of incidence matrices and via the structure of links. We show that rank-selected subcomplexes of saturated complexes are also saturated, and that order complexes of geometric lattices are saturated

On Modular Homology of Simplicial Complexes: Saturation

... homology, and these are the so-called saturated complexes. We show that certain conditions on the links of the complex imply saturation. We prove that Coxeter complexes and buildings are saturated

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate-Shafarevich group of an elliptic curve in the Z_p^2-extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.Comment: 33 pages, to appear in Journal of Algebr

Compact Grid Representation of Graphs

A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families