arXiv subjects
Publications and source records attributed to Jordi Guàrdia.
We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties $A_f$ attached by Shimura to normalized newforms $f \in S_2( Γ_0(N))$. We present all the curves corresponding to principally polarized surfaces $A_f$ for $N\leq500$.
We study monogenity of pure cubic number fields by means of Selmer groups of certain elliptic curves. A cubic number field with discriminant $D$ determines a unique nontrivial $\mathbb{F}_3$-orbit in the first cohomology group of the elliptic curve $E^D: y^2 = 4x^3 + D$ with respect to a certain 3-isogeny $ϕ$. Orbits corresponding to monogenic fields must lie in the soluble part of the Selmer group $S^ϕ(E^D/\mathbb{Q})$, and this gives a criterion to discard monogenity. From this, we can derive bounds on the number of monogenic cubic fields in terms of the rank of the elliptic curve. We can also determine the monogenity of many concrete pure cubic fields assuming GRH.
For a henselian valued field $(K,v)$ we establish a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaquié chains. This parallelism was only known for defectless irreducible polynomials.
Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaquié and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the factorization of $g$ over a henselization of $(K,v)$.
For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrowΛ$ of ordered groups, we analyze the structure of the tree formed by all $Λ$-valued extensions of $v$ to the polynomial ring $K[x]$. As an application, we find a model for the tree of all equivalence classes of valuations on $K[x]$ (without fixing their value group), whose restriction to $K$ is equivalent to $v$. In the henselian case, we apply these results to show that there is a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaquié chains. This parallelism was only known for defectless irreducible polynomials.
For a certain field $K$, we construct a valuation-algebraic valuation on the polynomial ring $K[x]$, whose Maclane--Vaquié chain consists of an infinite (countable) number of limit augmentations
For a prime $p$, the OM algorithm finds the $p$-adic factorization of an irreducible polynomial $f\in\mathbb{Z}[x]$ in polynomial time. This may be applied to construct $p$-integral bases in the number field $K$ defined by $f$. In this paper, we adapt the OM techniques to work with a positive integer $N$ instead of $p$. As an application, we obtain an algorithm to compute global integral bases in $K$, which does not require a previous factorization of the discriminant of $f$.
Let $ν$ be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field $K$. We prove comparison theorems between MacLane-Vaquié key polynomials for valuations $μ\leν$ and abstract key polynomials for $ν$. Also, some results on invariants attached to limit key polynomials are obtained. In particular, if $\operatorname{char}(K)=0$ we show that all limit key polynomials of unbounded continuous MacLane chains have numerical character equal to one.
We adapt an old local-to-global technique of Ore to compute, under certain mild assumptions, an integral basis of a number field without a previous factorization of the discriminant of the defining polynomial. In a first phase, the method yields as a by-product successive splittings of the discriminant. When this phase concludes, it requires a squarefree factorization of some base factors of the discriminant to terminate.
We present a formalization of geometric instruments that considers separately geometric and arithmetic aspects of them. We introduce the concept of tool, which formalizes a physical instrument as a set of axioms representing its geometric capabilities. We also define a map as a tool together with a set of points and curves as an initial reference. We rewrite known results using this new approach and give new relations between origami and other instruments, some obtained considering them as tools and others considering them as maps.
Let $(K,v)$ be a discrete valued field with valuation ring $\oo$, and let $\oo_v$ be the completion of $\oo$ with respect to the $v$-adic topology. In this paper we discuss the advantages of manipulating polynomials in $\oo_v[x]$ in a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the prime polynomial. These discrete parameters are a kind of DNA sequence common to all individuals in the same Okutsu class, and they contain relevant arithmetic information about the polynomial and the extension of $K_v$ that it determines.
Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work was recently reviewed and generalized by M. Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra of a valuation as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of the discrete valuations on $K(x)$ and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.