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Laia Amorós

Publications and source records attributed to Laia Amorós.

6 recordsLinked to original sources
arXiv2021

Even Unimodular Lattices from Quaternion Algebras

We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the $E_8$ lattice (resp. $E_8^2$ and $Λ_{16}$) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of $8$.

math.NT↗
arXiv2021

A machine learning approach to commutative algebra: Distinguishing table vs non-table ideals

We propose a novel approach to distinguish table vs non-table ideals by using different machine learning algorithms. We introduce the reader to table ideals, assuming some knowledge on commutative algebra and describe their main properties. We create a data set containing table and non-table ideals, and we use a feedforward neural network model, a decision tree and a graph neural networks for the classification. Our results indicate that there exists an algorithm to distinguish table ideals from non-table ideals, and we prove it along some novel results on table ideals.

math.AC↗
arXiv2020

Images of Galois representations in mod $p$ Hecke algebras

Let $(\mathbb{T}_f,\mathfrak{m}_f)$ denote the mod $p$ local Hecke algebra attached to a normalised Hecke eigenform $f$, which is a commutative algebra over some finite field $\mathbb{F}_q$ of characteristic $p$ and with residue field $\mathbb{F}_q$. By a result of Carayol we know that, if the residual Galois representation $\overlineρ_f:G_\mathbb{Q}\rightarrow\mathrm{GL}_2(\mathbb{F}_q)$ is absolutely irreducible, then one can attach to this algebra a Galois representation $ρ_f:G_\mathbb{Q}\rightarrow\mathrm{GL}_2(\mathbb{T}_f)$ that is a lift of $\overlineρ_f$. We will show how one can determine the image of $ρ_f$ under the assumptions that $(i)$ the image of the residual representation contains $\mathrm{SL}_2(\mathbb{F}_q)$, $(ii)$ that $\mathfrak{m}_f^2=0$ and $(iii)$ that the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow us to deduce the existence of certain $p$-elementary abelian extensions of big non-solvable number fields.

math.NT↗
arXiv2020

Well-Rounded Lattices: Towards Optimal Coset Codes for Gaussian and Fading Wiretap Channels

The design of lattice coset codes for wiretap channels is considered. Bounds on the eavesdropper's correct decoding probability and information leakage are first revisited. From these bounds, it is explicit that both the information leakage and error probability are controlled by the average flatness factor of the eavesdropper's lattice, which we further interpret geometrically. It is concluded that the minimization of the (average) flatness factor of the eavesdropper's lattice leads to the study of well-rounded lattices, which are shown to be among the optimal in order to achieve these minima. Constructions of some well-rounded lattices are also provided.

cs.IT↗
arXiv2017

Mumford curves covering p-adic Shimura curves and their fundamental domains

We give an explicit description of fundamental domains associated to the $p$-adic uniformisation of families of Shimura curves of discriminant $Dp$ and level $N\geq 1$, for which the one-sided ideal class number $h(D,N)$ is $1$. The obtained results generalise those in \cite[Ch. IX]{Gerritzen_vanderPut1980} for Shimura curves of discriminant $2p$ and level $N=1$. The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, $p$-adic good fundamental domains and their stable reduction-graphs. This is based on a detailed study of the modular arithmetic of an Eichler order of level $N$ inside the definite quaternion algebra of discriminant $D$, for which we generalise classical results of Hurwitz \cite{Hurwitz1896}. As an application, we prove general formulas for the reduction-graphs with lengths at $p$ of the considered families of Shimura curves.

math.NT↗