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Enric Nart - One of the best experts on this subject based on the ideXlab platform.

  • A New Computational Approach to Ideal Theory in Number Fields
    Foundations of Computational Mathematics, 2013
    Co-Authors: Jordi Guardia, Jesus Montes, Enric Nart
    Abstract:

    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 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 the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

  • a new computational approach to ideal theory in number fields
    arXiv: Number Theory, 2010
    Co-Authors: Jordi Guardia, Jesus Montes, Enric Nart
    Abstract:

    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.

Hui Zhu - One of the best experts on this subject based on the ideXlab platform.

  • group structures of elementary supersingular abelian varieties over finite fields
    Journal of Number Theory, 2000
    Co-Authors: Hui Zhu
    Abstract:

    Abstract Let A be a supersingular abelian variety over a finite field k which is k -isogenous to a power of a simple abelian variety over k . Write the characteristic Polynomial of the Frobenius endomorphism of A relative to k as f = g e for a Monic Irreducible Polynomial g and a positive integer e . We show that the group of k -rational points A ( k ) on A is isomorphic to ( Z / g (1)  Z ) e unless A 's simple component is of dimension 1 or 2, in which case we prove that A ( k ) is isomorphic to ( Z / g (1)  Z ) a ×( Z /( g (1)/2)  Z × Z /2 Z ) b for some non-negative integers a ,  b with a + b = e . In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A ( k )≅( Z / g (1)  Z ) e .

  • group structures of elementary supersingular abelian varieties over finite fields
    arXiv: Number Theory, 1998
    Co-Authors: Hui Zhu
    Abstract:

    Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic Polynomial f of the Frobenius endomorphism of A relative to k. If f=g^e for a Monic Irreducible Polynomial g and a positive integer e, we show that there is a group homomorphism A(k) --> (Z/g(1)Z)^e whose kernel and cokernel are elementary abelian 2-groups. In particular, this map is an isomorphism if the characteristic of k is 2 or A is simple of dimension greater than 2; in the last case one has e=1 or 2, and A(k) is isomorphic to (Z/g(1)Z)^e.

Jordi Guardia - One of the best experts on this subject based on the ideXlab platform.

  • A New Computational Approach to Ideal Theory in Number Fields
    Foundations of Computational Mathematics, 2013
    Co-Authors: Jordi Guardia, Jesus Montes, Enric Nart
    Abstract:

    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 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 the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

  • a new computational approach to ideal theory in number fields
    arXiv: Number Theory, 2010
    Co-Authors: Jordi Guardia, Jesus Montes, Enric Nart
    Abstract:

    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.

Maire Christian - One of the best experts on this subject based on the ideXlab platform.

  • Sur la séparation des caractères par les Frobenius
    2021
    Co-Authors: Euvrard Charlotte, Maire Christian
    Abstract:

    In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK. We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of factorization in Fp[X], where P ∈ Z[X] is a Monic, Irreducible Polynomial of squarefree discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group An over a fixed number field K. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two Irreducible unramified characters of the alterning group An,for n = 5, 7, 13

  • Sur la séparation des caractères par les Frobenius
    'Universitat Autonoma de Barcelona', 2017
    Co-Authors: Euvrard Charlotte, Maire Christian
    Abstract:

    In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK. We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of factorization in Fp[X], where P ∈ Z[X] is a Monic, Irreducible Polynomial of squarefree discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group An over a fixed number field K. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two Irreducible unramified characters of the alterning group An,for n = 5, 7, 13.In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK. We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of factorization in Fp[X], where P ∈ Z[X] is a Monic, Irreducible Polynomial of squarefree discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group An over a fixed number field K. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two Irreducible unramified characters of the alterning group An,for n = 5, 7, 13.In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK. We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of factorization in Fp[X], where P ∈ Z[X] is a Monic, Irreducible Polynomial of squarefree discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group An over a fixed number field K. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two Irreducible unramified characters of the alterning group An,for n = 5, 7, 13

Jesus Montes - One of the best experts on this subject based on the ideXlab platform.

  • A New Computational Approach to Ideal Theory in Number Fields
    Foundations of Computational Mathematics, 2013
    Co-Authors: Jordi Guardia, Jesus Montes, Enric Nart
    Abstract:

    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 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 the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

  • a new computational approach to ideal theory in number fields
    arXiv: Number Theory, 2010
    Co-Authors: Jordi Guardia, Jesus Montes, Enric Nart
    Abstract:

    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.