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Jordi Guardia

Publications and source records attributed to Jordi Guardia.

7 recordsLinked to original sources
arXiv2010

A new computational approach to ideal theory in number fields

Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher order of the defining equation $f(x)$. In this paper we show how to carry out the basic operations on fractional ideals of $K$ in terms of these constructive representations of the prime ideals. From a computational perspective, these results facilitate the manipulation of fractional ideals of $K$ avoiding two heavy tasks: the construction of the maximal order of $K$ and the factorization of the discriminant of $f(x)$. The main computational ingredient is Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

math.NT↗
arXiv2010

Okutsu invariants and Newton polygons

Let K be a local field of characteristic zero, O its ring of integers and F(x) a monic irreducible polynomial with coefficients in O. K. Okutsu attached to F(x) certain primitive divisor polynomials F_1(x),..., F_r(x), that are specially close to F(x) with respect to their degree. In this paper we characterize the Okutsu families [F_1,..., F_r] in terms of certain Newton polygons of higher order, and we derive some applications: closed formulas for certain Okutsu invariants, the discovery of new Okutsu invariants, or the construction of Montes approximations to F(x); these are monic irreducible polynomials sufficiently close to F(x) to share all its Okutsu invariants. This perspective widens the scope of applications of Montes' algorithm, which can be reinterpreted as a tool to compute the Okutsu polynomials and a Montes approximation, for each irreducible factor of a monic separable polynomial f(x) in O[x].

math.NT↗
arXiv2008

Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer.

math.NT↗
arXiv2008

Newton polygons of higher order in algebraic number theory

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by Ø. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.

math.NT↗
arXiv2008

Genus two curves with quaternionic multiplication and modular jacobian

We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$.

math.NT↗
arXiv2004

Abelian surfaces of GL2-type as Jacobians of curves

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian varieties are isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.

math.NT↗