arXiv subjects

Guillermo Mantilla-Soler

Publications and source records attributed to Guillermo Mantilla-Soler.

At least 19 recordsLinked to original sources

arXiv2024

Dense generic well-rounded lattices

It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice $\mathbb{Z}^n$ is the simplest example. The integer lattice is also an example of a generic well-rounded lattice, i.e., a well-rounded lattice with a minimal kissing number. However, the integer lattice has the worst density among well-rounded lattices. In this paper, the problem of constructing explicit generic well-rounded lattices with dense sphere packings is considered. To this end, so-called tame lattices recently introduced by Damir and Mantilla-Soler are utilized. Tame lattices came to be as a generalization of the ring of integers of certain abelian number fields. The sublattices of tame lattices constructed in this paper are shown to always result in either a generic well-rounded lattice or the lattice $A_n$, with density ranging between that of $\mathbb{Z}^n$ and $A_n$. In order to find generic well-rounded lattices with densities beyond that of $A_n$, explicit deformations of some known densest lattice packings are constructed, yielding a family of generic well-rounded lattices with densities arbitrarily close to the optimum. In addition to being an interesting mathematical problem on its own right, the constructions are also motivated from a more practical point of view. Namely, generic well-rounded lattices with high packing density make good candidates for lattice codes used in secure wireless communications.

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arXiv2022

The integral trace form as a complete invariant for real $S_n$ number fields

In the past the first named author has studied to what extent the integral trace can characterize a number field beyond what the discriminant does. The cases of cyclic number fields and non-totally real fields are more or less settled, concluding that for such fields the integral trace does not always characterize the field. In this paper we show that the integral trace is a complete invariant for degree $n$, $S_n$ real number number fields that satisfy certain ramification bound. Among the real $S_n$ fields that our results cover, there are those of square free different ideal. Moreover, for such fields we find an explicit description of the isometry group of the integral trace.

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arXiv2022

The Shape of cyclic number fields

Let $m>1$ and $\mathfrak{d} \neq 0$ be integers such that $v_{p}(\mathfrak{d}) \neq m$ for any prime $p$. We construct a matrix $A(\mathfrak{d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak{d}$ with the following property: For any tame $\mathbb{Z}/m\mathbb{Z}$-number field $K$ of discriminant $\mathfrak{d}$ the matrix $A(\mathfrak{d})$ represents the Gram matrix of the integral trace zero form of $K$. In particular, we have that the integral trace zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant

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arXiv2022

Bases of minimal vectors in tame lattices

Motivated by the behavior of the trace pairing over tame cyclic number fields, we introduce the notion of tame lattices. Given an arbitrary non-trivial lattice $\mathcal{L}$ we construct a parametric family of full-rank sub-lattices $\{\mathcal{L}_α\}$ of $\mathcal{L}$ such that whenever $\mathcal{L}$ is tame each $\mathcal{L}_α$ has a basis of minimal vectors. Furthermore, for each $\mathcal{L}_α$ in the family a basis of minimal vectors is explicitly constructed.

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arXiv2021

Density questions on arithmetic equivalence

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime $p$ is the same in both fields for almost all prime $p$. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely in the degree of the fields. More specifically, for every positive $n$ we exhibit a positive constan $c_{n}$ such that any two degree $n$ number fields $K$ and $L$ are arithmetically equivalent if and only if the set of primes $p$ such that the arithmetic type of $p$ in $K$ and $L$ is not the same has Dirichlet density at most $c_n$. We in fact show that $\displaystyle c_n=\frac{1}{4n^2}$ works and give a heuristic evidence that points to the fact that this value might be improved to $\displaystyle \frac{2}{n^2}$. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such number.

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arXiv2020

A proof of a conjecture on trace-zero forms and shapes of number fields

In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog statement for cubic fields which was proved using Bhargava's higher composition laws on cubes. Here, using Bhargava's parametrization of quartic fields we prove the conjecture by generalizing the ideas used in the cubic case. Since at the moment, for arbitrary degrees, there is nothing like Bhargava's parametrizations we cannot deal with degrees $n > 5$ in a similar fashion. Nevertheless, using some of our previous work on trace forms we generalize this result to higher degrees; we show that if $n \ge 3$ is an integer such that $(\mathbb{Z}/n\mathbb{Z})^{*}$ is a cyclic group, then the shape is a complete invariant for totally real degree $n$ number fields with fundamental discriminant.

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arXiv2020

The trace form over cyclic number fields

In the mid 80's Conner and Perlis showed that for cyclic number fields of prime degree $p$ the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.

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arXiv2019

Theta series and number fields: theorems and experiments

We construct certain $θ$-series associated to number fields and prove that for number fields of degree less than equal to 4, these $θ$-series are number field invariants. We also investigate whether or not the collection of $θ$-series associated to number fields of the same degree and discriminant are linearly independent. This is known to be true if the degree of the number field is less than or equal to 3. We do not prove in this paper that they are linearly independent in general but we do give computational and heuristic evidence that we would expect them to be.

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arXiv2019

An introduction to $Γ$-number fields

It follows from generalities of quadratic forms that the spinor class of the integral trace of a number field determines the signature and the discriminant of the field. In this paper we define a family of number fields, that contains among others all odd degree Galois tame number fields, for which the converse is true. In other words, for a number field $K$ in such family we prove that the spinor class of the integral trace carries no more information about $K$ than the determinant and the signature do.

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arXiv2019

The $(α, β)-$ramification invariants of a number field

Let $L$ be a number field. For a given prime $p$ we define integers $α_{p}^{L}$ and $β_{p}^{L}$ with some interesting arithmetic properties. For instance, $β_{p}^{L}$ is equal to $1$ whenever $p$ does not ramify in $L$ and $α_{p}^{L}$ is divisible by $p$ whenever $p$ is wildly ramified in $L$. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of $L$. Moreover, if the residue class mod $p$ of $α_{p}^{L}$ is not zero for all $p$ then such residues determine the genus of the integral trace.

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arXiv2019

On a question of Perlis and Stuart regarding arithmetic equivalence

Let $K$ be a number field. The $K$-arithmetic type of a rational prime $\ell$ is the tuple $A_{K}(\ell)=(f^{K}_{1},...,f^{K}_{g_{\ell}})$ of the residue degrees of $\ell$ in $K$, written in ascending order. A well known result of Perlis from the 70's states that two number fields have the same Dedekind zeta function if and only if for almost all primes $\ell$ the arithmetic types of $\ell$ in both fields coincide. By the end of the 90's Perlis and Stuart asked if having the same zeta function implies that for ramified primes the sum of the ramification degrees coincide. Here we study and answer their question for septic number fields.

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arXiv2019

On a characterization of path connected topological fields

The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space $X$ and the space of maximal ideals of the ring of real valued continuous functions $C(X,\mathbb{R})$. More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if $F$ is a path connected topological field.

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arXiv2018

An $\ell-p$ switch trick to obtain a new elementary proof of a criterion for arithmetic equivalence

Two number fields are called arithmetically equivalent if they have the same Dedekind zeta function. In the 1970's Perlis showed that this is equivalent to the condition that for almost every rational prime $\ell$ the arithmetic type of $\ell$ is the same in each field. In the 1990's Perlis and Stuart gave an unexpected characterization for arithmetic equivalence; they showed that to be arithmetically equivalent it is enough for almost every prime $\ell$ to have the same number of prime factors in each field. Here, using an $\ell-p$ switch trick, we provide an elementary proof of that fact based on a classical result of Smith from the 1870's.

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arXiv2015

On the Arithmetic determination of the trace

Let $K$ be a number field, which is tame and non totally real. In this article we give a numerical criterion, depending only on the ramification behavior of ramified primes in $K$, to decide whether or not the integral trace of $K$ is isometric to the integral trace of another number field $L$. As a byproduct of our proofs here, and in contrast with our previous results for cubic fields of positive discriminant, we show that for cubic fields of negative discriminant isometry between integral traces is equivalent to equality of discriminants.

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arXiv2015

The shape of $\mathbb{Z}/\ell\mathbb{Z}$-number fields

Let $\ell$ be a prime and let $L/\mathbb{Q}$ be a Galois number field with Galois group isomorphic to $\mathbb{Z}/\ell\mathbb{Z}$. We show that the {\it shape} of $L$ is either $\frac{1}{2}\mathbb{A}_{\ell-1}$ or a fixed sub lattice depending only on $\ell$; such a dichotomy in the value of the shape only depends on the type of ramification of $L$. This work is motivated by a result of Bhargava and Shnidman, and a previous work of the first named author, on the shape of $\mathbb{Z}/3\mathbb{Z}$ number fields.

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arXiv2015

The spinor genus of the integral trace

Let $K$ be a number field of degree at least $3$. In this article we show that the genus of the integral trace form of $K$ contains only one spinor genus. Additionally we show that exactly $43%$ (resp. $29%$, resp. $58%$) of quadratic (resp. real quadratic, resp. imaginary quadratic) fields have the same property.

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arXiv2014

Weak arithmetic equivalence

Inspired by the invariant of a number field given by its zeta function, we define the notion of {\it weak arithmetic equivalence}, and show that under certain ramification hypothesis, this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under weak arithmetic equivalence.

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arXiv2013

A space of weight one modular forms attached to totally real cubic number fields

Let $d$ be a positive fundamental discriminant, and let $\mathcal{C}_{d}$ be the set of isomorphism classes of cubic number fields of discriminant $d$. For each $K \in \mathcal{C}_{d}$, we construct a weight 1 modular form $f_{K}$ with level $3^{\pm 1}d$ and nebentypus $\left( \frac{-3^{\pm 1}d}{\cdot} \right)$. We show that the form $f_{K}$ completely determines the field $K$. Moreover, we show that $\{f_{K} : K \in \mathcal{C}_{d}\}$ is a linearly independent set.

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