Covolumes of nonuniform lattices in PU(n, 1)

This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conjugacy) in PU(9, 1) of smallest Euler--Poincar\'e characteristic amongst all nonuniform arithmetic lattices in PU(n, 1). We also show that for each even n, there are arbitrarily large families of nonisomorphic maximal nonuniform lattices in PU(n, 1) of equal covolume.Comment: To appear in American Journal of Mathematic

A Note on Optimal Unimodular Lattices

The highest possible minimal norm of a unimodular lattice is determined in dimensions n <= 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8*10^20 in dimension 33). Unimodular lattices with no roots exist if and only if n >= 23, n not = 25.Comment: 8 page

A Sufficient Condition for J\'onsson's Conjecture and its Relationship with Finite Semidistributive lattices

This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable lattices) are sublattices of a free lattice?" Despite partial progress over the decades, the problem is still unsolved. There is emphasis on the countable case because the current body of knowledge on sublattices of free lattices is most concentrated on when these sublattices are countably infinite. It is known that sublattices of free lattices which are finite can be characterized as being those lattices which satisfy Whitman's condition and are semidistributive. This assertion was conjectured by B. Jonsson in the 1960's and proven by J.B. Nation in 1980. However, there is a desire for a new proof to this deep result as Nation's proof is very involved and more insight into sublattices of free lattices is sought after. In this article, a sufficient condition involving a construct known as a join minimal pair, or just a minimal pair, implying J'onsson's conjecture is proven. Minimal pairs were first defined by H. Gaskill when analysing sharply transferable lattices. Using this sufficient condition, research by I.Rival and B.Sands is used to compare this condition with properties of finite semidistributive lattices and in the process refute the main assertion of a manuscript by H.Muhle. Moreover, inspired by the approaches used by Henri Muhle, I will make a partial result which investigates a possible forbidden sublattice characterization involving breadth-two planar semidistributive lattices. To the best of my knowledge, the two results of this article (the sufficient condition for Jonsson's conjecture and the partial result aforementioned) are new

Lattices of minimal covolume in SL_n(R)

The objective of this paper is to determine the lattices of minimal covolume in SL_n(R), for n greater than 3. The answer turns out to be the simplest one: SL_n(Z) is, up to automorphism, the unique lattice of minimal covolume in SL_n(R). In particular, lattices of minimal covolume in SL_n(R) are non-uniform when n is greater than 3, contrasting with Siegel's result for SL_2(R). This answers for SL_n(R) the question of Lubotzky: is a lattice of minimal covolume typically uniform or not?Comment: 28 pages, 1 figure, 6 table

Finite atomic lattices and resolutions of monomial ideals

In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be realized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cellular structure of a minimal free resolution of a monomial ideal M can be extended to minimal resolutions of certain monomial ideals whose LCM lattices are greater than that of M in L(n).Comment: 22 pages, 2 figure

A Bound on the Norm of Shortest Vectors in Lattices Arising from CM Number Fields

This paper partially addresses the problem of characterizing the lengths of vectors in a family of Euclidean lattices that arise from any CM number field. We define a modified quadratic form on these lattices, the weighted norm, that contains the standard field trace as a special case. Using this modified quadratic form, we obtain a bound on the field norm of any vector that has a minimal length in any of these lattices, in terms of a basis for the group of units of the ring of integers of the field. For any CM number field F, we prove that there exists a finite set of elements of F which allows one to find the set of minimal vectors in every principal ideal of the ring of integers of F. We interpret our result in terms of the asymptotic behavior of a Hilbert modular form, and consider some of the computational implications of our theorem. Additionally, we show how our result can be applied to the specific Craig's Difference Lattice problem, which asks us to find the minimal vectors in lattices arising from cyclotomic number fields.Comment: 15 page

Spherical 2-designs and lattices from Abelian groups

We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of the automorphism groups of these lattices, strengthening our previous results in this direction.Comment: 12 page

Modular Lattices from a Variation of Construction A over Number Fields

We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited

Quasicomplemented residuated lattices

In this paper, the class of quasicomplemented residuated lattices is introduced and investigated, as a subclass of residuated lattices in which any prime filter not containing any dense element is a minimal prime filter. The notion of disjunctive residuated lattices is introduced and it is observed that a residuated lattice is Boolean if and only if it is disjunctive and quasicomplemented. Finally, some characterizations for quasicomplemented residuated lattices are given by means of the new notion of $\alpha$-filters.Comment: arXiv admin note: text overlap with arXiv:1812.11511, arXiv:1812.1151

Completeness of Unbounded Convergences

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence.Comment: 15 page