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79 results on '"Henri Cohen"'

1. On Dℓ‐extensions of odd prime degree ℓ

Author

Frank Thorne and Henri Cohen

Subjects
Pure mathematics, Conjecture, Mathematics::Number Theory, General Mathematics, 010102 general mathematics, Prime degree, Generating function, Binary number, 010103 numerical & computational mathematics, 01 natural sciences, symbols.namesake, Quadratic equation, symbols, 0101 mathematics, Dirichlet series, Mathematics, Resolvent
Abstract

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_l$-extensions of odd prime degree $l$ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics.

Published
2020

2. Computational Number Theory in Relation with L-Functions

Author

Henri Cohen, Lithe and fast algorithmic number theory (LFANT), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Ilker Inam, Engin Büyükaşık, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Gauss sums, Computation, Numerical analysis, 010102 general mathematics, Gauss, Algebraic variety, 01 natural sciences, Dirichlet distribution, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Hypergeometric motives, symbols.namesake, Finite field, L-functions, Algebraic varieties, Gauss sum, 0103 physical sciences, symbols, Applied mathematics, 010307 mathematical physics, 0101 mathematics, Computational number theory, Mathematics
Abstract

International audience; We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance Dirichlet L-functions, involving in particular the study of inverse Mellin transforms); we also give a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions.

Published
2019

3. An Introduction to Modular Forms

Author

Henri Cohen, Lithe and fast algorithmic number theory (LFANT), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Ilker Inam, Engin Büyükaşık, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Algebra, Classical theory, 4. Education, Modular form, Modular forms, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Mathematics, Style (sociolinguistics)
Abstract

In this course, we introduce the main notions relative to the classical theory of modular forms. A complete treatise in a similar style can be found in the author’s book joint with Stromberg (Cohen and Stromberg, Modular Forms: A Classical Approach, Graduate Studies in Math. 179, American Math. Soc. (2017) [1]).

Published
2019

4. Expansions at Cusps and Petersson Products in Pari/GP

Author

Henri Cohen, Lithe and fast algorithmic number theory (LFANT), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Johannes Blümlein, Carsten Schneider, Peter Paule, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Current (mathematics), Mathematics::Number Theory, 010102 general mathematics, Modular form, Modular forms, 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], symbols.namesake, Fourier transform, Eisenstein series, symbols, 0101 mathematics, Petersson products, Mathematics
Abstract

International audience; We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov-Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we then give a number of methods for computing arbitrary Petersson products. All this is available in the current release of the Pari/GP package.

Published
2019

5. Modular Forms in Pari/GP

Author

Henri Cohen, Karim Belabas, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Mathematics - Number Theory, Applied Mathematics, Mathematics::Number Theory, Modular form, Scalar (mathematics), 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Theoretical Computer Science, 010101 applied mathematics, Algebra, Computational Mathematics, symbols.namesake, Mathematics (miscellaneous), Fourier transform, Eisenstein series, FOS: Mathematics, symbols, Number Theory (math.NT), 0101 mathematics, Complex plane, Mathematics
Abstract

We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of non-eigenforms, and compute general Petersson scalar products., Research in the Mathematical Sciences (RMSS), In press

Published
2018

6. Identities for field extensions generalizing the Ohno–Nakagawa relations

Author

Simon Rubinstein-Salzedo, Frank Thorne, and Henri Cohen

Subjects
Pure mathematics, Algebra and Number Theory, Prehomogeneous vector space, Mathematics - Number Theory, Degree (graph theory), Mathematics::Number Theory, Galois group, Binary number, Prime (order theory), Identity (mathematics), Field extension, FOS: Mathematics, Number Theory (math.NT), Reflection principle, Mathematics
Abstract

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of `extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper we generalize their result by proving a similar identity relating certain degree l fields with Galois groups D_l and F_l respectively, for any odd prime l, and in particular we give another proof of the Ohno-Nakagawa relation without appealing to binary cubic forms., Version 2, 16 pages, to appear in Compositio

Published
2015

8. Vanishing and non-vanishing theta values

Author

Henri Cohen, Don Zagier, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Max-Planck-Institut für Mathematik (MPI), European Project: 278537,EC:FP7:ERC,ERC-2011-StG_20101014,ANTICS(2012), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Pure mathematics, Series (mathematics), Root of unity, General Mathematics, 010102 general mathematics, Modular form, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Dirichlet character, Ramanujan's sum, Combinatorics, symbols.namesake, Number theory, Gauss sum, 0103 physical sciences, symbols, 010307 mathematical physics, 0101 mathematics, Algebraic number, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

For a primitive Dirichlet character $$\chi $$ of conductor $$N$$ set $$\begin{aligned} \Theta (\chi )=\sum _{n\in \mathbb Z } n^\epsilon \chi (n)\,e^{-\pi n^2/N} \end{aligned}$$ (where $$\epsilon =0$$ for $$\chi $$ even, $$\epsilon =1$$ for $$\chi $$ odd), the value of the associated theta series at its point of symmetry under the modular transformation $$\tau \rightarrow -1/\tau $$ . These numbers are related by $$\Theta (\chi )=W(\chi )\Theta (\bar{\chi })$$ to the root number of the $$L$$ -series of $$\chi $$ and hence can be used to calculate the latter quickly if they do not vanish. We describe experiments showing that $$\Theta (\chi )\ne 0$$ for all $$\chi $$ with $$N\le 52{,}100$$ (roughly 500 million primitive characters) except for precisely two characters (up to $$\chi \rightarrow \bar{\chi }$$ ), of conductors $$300$$ and $$600$$ . The proof that $$\Theta (\chi )$$ vanishes in these two cases uses properties of Ramanujan’s modular function of level $$5$$ . We also characterize all $$\chi $$ for which $$W(\chi )$$ is a root of unity and describe some experimental results concerning the algebraic numbers $$\Theta (\chi )/\eta (i)^{1+2\epsilon }$$ when $$N$$ is prime.

Published
2013

10. Explicit Methods in Number Theory

Author

Don Zagier, Henri Cohen, and Hendrik W. Lenstra

Subjects
Algebra, Class (set theory), Range (mathematics), Finite field, Number theory, Rational point, Modular form, General Medicine, Integer factorization, Mathematics
Abstract

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics included modular forms, va- rieties over finite fields, rational and integral points on varieties, class groups, and integer factorization.

Published
2009

11. Exact counting of $D_\ell$ number fields with given quadratic resolvent

Author

Henri Cohen, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), European Project: 278537,EC:FP7:ERC,ERC-2011-StG_20101014,ANTICS(2012), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Discrete mathematics, Algebra and Number Theory, Applied Mathematics, 010102 general mathematics, Algebraic number field, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Computational Mathematics, Quadratic equation, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Resolvent
Abstract

International audience

Published
2015

12. Counting discriminants of number fields

Author

Henri Cohen, Michel Olivier, and Francisco Diaz y Diaz

Subjects
Combinatorics, Differential Galois theory, Embedding problem, Algebra and Number Theory, Galois cohomology, Mathematics::Number Theory, Galois group, Permutation group, Algebraic number field, Galois module, Mathematics, Resolvent
Abstract

For each transitive permutation group G on n letters with n 4, we give without proof results, conjectures, and numer- ical computations on discriminants of number fields L of degree n over Q such that the Galois group of the Galois closure of L is isomorphic to G.

Published
2006

13. Counting cyclic quartic extensions of a number field

Author

Michel Olivier, Francisco Diaz y Diaz, and Henri Cohen

Subjects
Pure mathematics, Algebra and Number Theory, Quartic function, Calculus, Algebraic number field, Mathematics
Abstract

Nous donnons des formules asymptotiques pour le nombre d'extensions cycliques quartiques d'un corps de nombres general.

Published
2005

14. Analysis of the Sliding Window Powering Algorithm

Author

Henri Cohen

Subjects
Elliptic curve, Exponentiation, Number theory, Addition chain, Applied Mathematics, Sliding window protocol, Prime field, Algorithm, Software, Computer Science Applications, Mathematics
Abstract

We analyze precisely the parameters entering the sliding window powering method, and apply this to the case of an elliptic curve over a large finite prime field.

Published
2004

15. Cyclotomic extensions of number fields

Author

Michel Olivier, Francisco Diaz y Diaz, and Henri Cohen

Subjects
Discrete mathematics, Pure mathematics, Mathematics(all), Almost prime, General Mathematics, Prime ideal, Mathematics::Number Theory, Prime number, Splitting of prime ideals in Galois extensions, Prime element, Prime k-tuple, Associated prime, Prime power, Mathematics
Abstract

Let K be a number field, l a prime number, ζ l a primitive l -th root of unity and Kz = K(ζ l ). In this paper, we first give a detailed description of the discriminant, conductor, different and prime ideal decomposition of the extension Kz/K. We apply this to obtain the Galois-module structure of certain finite modules associated to prime ideals above l , and we also give the Galois-module structure of the unit group of Kz modulo l th powers.

Published
2003
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16. Enumerating quartic dihedral extensions of ${\Bbb Q}$ with signatures

Author

Henri Cohen

Subjects
Combinatorics, Algebra and Number Theory, Number theory, Quartic function, Quartic reciprocity, Geometry and Topology, Isomorphism, Dihedral angle, Algebraic number field, Signature (topology), Mathematics
Abstract

Dans un precedent article, nous avons donne des formules asymptotiques pour le nombre de classes d'isomorphismes d'extensions D 4 classees par discriminant croissant, que la signature des extensions soit ou non specifiee. Nous avons egalement donne des formules exactes tres efficaces pour ce nombre quand on ne specifie pas la signature. Le but du present article est de donner de telles formules exactes quand la signature est imposee. Le probleme se complique du fait de l'apparition de caracteres sur les groupes de classes de rayon qui ne sont pas des caracteres de genre.

Published
2003

17. Constructing complete tables of quartic fields using Kummer theory

Author

Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier

Subjects
Pure mathematics, Algebra and Number Theory, Kummer theory, Geometry of numbers, Applied Mathematics, Computation, Algebraic number field, Table (information), Algebra, Computational Mathematics, Number theory, Discriminant, Quartic function, Mathematics
Abstract

We explain how to construct tables of quartic fields of discriminant less than or equal to a given bound in an efficient manner using Kummer theory, instead of the traditional (and much less efficient) method using the geometry of numbers. As an application, we describe the computation of quartic fields of discriminant up to 10 7 10^7 , the corresponding table being available by anonymous ftp.

Published
2002

18. [Untitled]

Author

Francisco Diaz Y Diaz, Michel Olivier, and Henri Cohen

Subjects
Combinatorics, symbols.namesake, Algebra and Number Theory, Discriminant, Series (mathematics), Quartic function, Generating function, symbols, Asymptotic formula, Isomorphism, Quartic surface, Dirichlet series, Mathematics
Abstract

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of ℚ. From this series we deduce an asymptotic formula for the number of isomorphism classes of such quartic extensions with discriminant up to a given bound. On the other hand, by using essentially classical results of genus theory combined with elementary analytical methods such as the method of the hyperbola, we show how to compute exactly this number up to quite large bounds, and we give a table of selected values.

Published
2002

19. Dirichlet series associated to cubic fields with given quadratic resolvent

Author

Frank Thorne, Henri Cohen, Lithe and fast algorithmic number theory (LFANT), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Columbia], University of South Carolina [Columbia], European Project: 278537,EC:FP7:ERC,ERC-2011-StG_20101014,ANTICS(2012), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Class (set theory), Mathematics - Number Theory, General Mathematics, Carry (arithmetic), 010102 general mathematics, 11R29, Field (mathematics), 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], symbols.namesake, Quadratic equation, 11Y40, 11R37, Quartic function, 11R16, FOS: Mathematics, symbols, Quadratic field, Number Theory (math.NT), 0101 mathematics, Dirichlet series, Resolvent, Mathematics
Abstract

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q. In a companion paper we do the same for quartic fields having a given cubic resolvent. As an application (not present in the initial version), we compute tables of the number of S_3-sextic fields E with |Disc(E)| < X, for X ranging up to 10^23. An accompanying PARI/GP implementation is available from the second author's website., 16 pages, submitted. Revised version: includes counts of S_3-sextic fields

Published
2014

20. Algorithmic Methods for Finitely Generated Abelian Groups

Author

Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier

Subjects
Discrete mathematics, Computational Mathematics, Torsion subgroup, Algebra and Number Theory, Elementary abelian group, Abelian category, Abelian group, Rank of an abelian group, Free abelian group, Non-abelian group, Arithmetic of abelian varieties, Mathematics
Abstract

We describe algorithmic tools to compute with exact sequences of Abelian groups. Although simple in nature, these are essential for a number of applications such as the determination of the structure of (ZK/m)*for an ideal m of a number field K, of ray class groups of number fields, and of conductors of the corresponding Abelian extensions.

Published
2001
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21. Haberland's formula and numerical computation of Petersson scalar products

Author

Henri Cohen, Lithe and fast algorithmic number theory (LFANT), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Everett W. Howe and Kiran S. Kedlaya, European Project: 278537,EC:FP7:ERC,ERC-2011-StG_20101014,ANTICS(2012), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Algebra, Mathematics::Group Theory, Mathematics::Number Theory, Computation, 010102 general mathematics, Scalar (mathematics), 010103 numerical & computational mathematics, 0101 mathematics, 16. Peace & justice, Mathematics::Geometric Topology, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Mathematics
Abstract

International audience; We study several methods for the numerical computation of Petersson scalar products, and in particular we prove a generalization of Haberland's formula to any subgroup of finite index G of Gamma = PSL_2 (Z), which gives a fast method to compute these scalar products when a Hecke eigenbasis is not necessarily available.

Published
2013
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22. Densité des discriminants des extensions cycliques de degré premier

Author

Francisco Diaz y Diaz, Henri Cohen, and Michel Olivier

Subjects
General Medicine, Humanities, Mathematics
Abstract

Resume Nous donnons une formule asymptotique pour le nombre d'extensions relatives cycliques de degre premier d'un corps de nombres.

Published
2000

23. Comptage exact de discriminants d'extensions abéliennes

Author

Henri Cohen

Subjects
Combinatorics, symbols.namesake, Algebra and Number Theory, Number theory, symbols, Abelian extension, Transitive group, Isomorphism class, Möbius function, Dirichlet series, Group theory, Convolution, Mathematics
Abstract

Le but de cet article est d'expliquer comment calculer exactement le nombre de classes d'isomorphismes d'extensions abeliennes de Q en degre inferieur ou egal a 4 et de discriminant majore par une borne donnee. On parvient par exemple a calculer le nombre de corps cubiques cycliques de discriminant inferieur ou egal a 10 37 .

Published
2000

24. Tables of octic fields with a quartic subfield

Author

Michel Olivier, F. Diaz y Diaz, and Henri Cohen

Subjects
Pure mathematics, Algebra and Number Theory, Degree (graph theory), Mathematics::Number Theory, Applied Mathematics, Galois group, Algebraic number field, Algebra, Computational Mathematics, Number theory, Discriminant, Quartic function, Class field theory, Dedekind zeta function, Mathematics
Abstract

We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.

Published
1999

25. A survey of computational class field theory

Author

Henri Cohen

Subjects
Pure mathematics, Algebra and Number Theory, Kummer theory, Number theory, Class field theory, Complex multiplication, Abelian group, Mathematics
Published
1999

26. Computing ray class groups, conductors and discriminants

Author

F. Diaz y Diaz, Michel Olivier, and Henri Cohen

Subjects
Computational Mathematics, Pure mathematics, Class (set theory), Algebra and Number Theory, Number theory, Applied Mathematics, Computation, Structure (category theory), Ideal (ring theory), Algebraic number field, Abelian group, Group theory, Mathematics
Abstract

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of ( Z K / m ) ∗ (\mathbb {Z}_{K}/\mathfrak {m})^{*} for an ideal m \mathfrak {m} of a number field K K , as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

Published
1998

27. Hermite and Smith normal form algorithms over Dedekind domains

Author

Henri Cohen

Subjects
Algebra and Number Theory, Hermite polynomials, Applied Mathematics, Dedekind sum, Dedekind domain, Algebraic number field, Hermite normal form, Computational Mathematics, symbols.namesake, symbols, Dedekind eta function, Dedekind cut, Algorithm, Mathematics, Smith normal form
Abstract

We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields.

Published
1996

28. Counting Cubic Extensions with given Quadratic Resolvent

Author

Henri Cohen, Anna Morra, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Kummer theory, Fundamental theorem of Galois theory, Mathematics::Number Theory, 010103 numerical & computational mathematics, Discriminant counting, Cubic extensions, 01 natural sciences, Combinatorics, symbols.namesake, Quartic function, FOS: Mathematics, Artin–Schreier theory, Cubic field, Number Theory (math.NT), 0101 mathematics, Mathematics, Resolvent, Algebra and Number Theory, Mathematics - Number Theory, 010102 general mathematics, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Discriminant, symbols, Quadratic field, 11R20
Abstract

Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha})$, for an explicit $\alpha, Comment: 19 pages

Published
2010
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29. Elementary 3-descent with a 3-isogeny

Author

Fabien Pazuki, Henri Cohen, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Sciences et Technologies - Bordeaux 1, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Sciences et Technologies - Bordeaux 1 (UB)

Subjects
Isogeny, Algebra and Number Theory, Mathematics - Number Theory, Computation, 010102 general mathematics, Mathematics::History and Overview, Rank of an elliptic curve, 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Interpretation (model theory), Algebra, 11G05, 14G05, 14H52, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Mathematics, Descent (mathematics)
Abstract

International audience; In this expository paper, we show how to use in practice 3-descent with a 3-isogeny to find an estimate for the rank of an elliptic curve having a rational 3-torsion subgroup, and we also give a geometric interpretation of these computations.

Published
2009

30. A polynomial reduction algorithm

Author

Henri Cohen and Francisco Diaz y Diaz

Subjects
Reduction (complexity), Combinatorics, Reciprocal polynomial, Polynomial, Jenkins–Traub algorithm, Degree of a polynomial, Chebyshev nodes, Matrix polynomial, Mathematics, Wilkinson's polynomial
Published
1991

31. Raabes formula for p-adic gamma and zeta functions

Author

Eduardo Friedman and Henri Cohen

Subjects
Pure mathematics, symbols.namesake, Algebra and Number Theory, Functional equation, Mathematical analysis, symbols, Geometry and Topology, L-function, Gamma function, Mathematics, Riemann zeta function
Abstract

La formule de Raabe classique donne la valeur de l'integrale de la fonction log gamma d'Euler sur un intervalle de longueur 1. Nous calculons des integrales p-adiques analogues pour les fonctions log gamma p-adiques de Diamond et de Morita, et nous montrons que chacune de ces fonctions est caracterisee de maniere unique par son equation fonctionnelle et sa formule de Raabe p-adique. Nous demontrons aussi une formule de type Raabe pour les fonctions zeta de Hurwitz p-adiques.

Published
2008

32. Explicit estimates for summatory functions linked to the Möbius $\mu$-function

Author

Henri Cohen, Francois Dress, and Mahomed El Marraki

Subjects
General Mathematics, 11Y35, Square-free integer, Function (mathematics), summatory functions, Term (logic), Type (model theory), Möbius function, Combinatorics, 11N37, 11Y70, Remainder, Mathematics
Abstract

Let $M(x)$ be the summatory function of the Möbius function and $R(x)$ be the remainder term for the number of squarefree integers up to $x$. In this paper, we prove the explicit bounds $|M(x)

Published
2007

34. A Survey of Discriminant Counting

Author

Francisco Diaz y Diaz, Henri Cohen, and Michel Olivier

Subjects
Combinatorics, Discrete mathematics, Number theory, Discriminant, Enumeration, Bibliography, Algebraic number field, Mathematical proof, Mathematics
Abstract

We give a survey of known results on the asymptotic and exact enumeration of discriminants of number fields, both in the absolute and relative case. We give no proofs, and refer instead to the bibliography.

Published
2002

35. Computational Aspects of Number Theory

Author

Henri Cohen

Subjects
Number theory, Mathematics education, Mathematics
Abstract

Number theory is one of the oldest branches of mathematics, and numbertheoretical results can be found in most ancient cultures. Like many other subjects, it has flourished into a major mathematical discipline. However, it has properties that make it almost unique compared to other mathematical subjects.

Published
2001

36. Fundamental Results and Algorithms in Dedekind Domains

Author

Henri Cohen

Subjects
Discrete mathematics, Prüfer domain, Prime ideal, Projective module, Free module, Euclidean domain, Principal ideal domain, Algebraic number field, Ring of integers, Mathematics
Abstract

The easiest way to start studying number fields is to consider them per se, as absolute extensions of √; this is, for example, what we have done in [Coh0]. In practice, however, number fields are frequently not given in this way. One of the most common other ways is to give a number field as a relative extension, in other words as an algebra L/K over some base field K that is not necessarily equal to √. necessarily equal to ℚ. In this case, the basic algebraic objects such as the ring of integers ℤ L and the ideals of ℤ L , are not only ℤ-modules, but also ℤ K- modules. The ℤ K -module structure is much richer and must be preserved. No matter what means are chosen to compute ℤ L , we have the problem of representing the result. Indeed, here we have a basic stumbling block: considered as ℤ-modules, ℤ L or ideals of ℤ L are free and hence may be represented by ℤ-bases, for instance using the Hermite normal form (HNF); see, for example, [Coh0, Chapter 2]. This theory can easily be generalized by replacing ℤ with any other explicitly computable Euclidean domain and, under certain additional conditions, to a principal ideal domain (PID). In general, ℤ K is not a PID, however, and hence there is no reason for ℤ L to be a free module over ℤ K- A simple example is given by K = ℚ(√-10) and L = K(√1) (see Exercise 22 of Chapter 2).

Published
2000

37. Computing Defining Polynomials Using Kummer Theory

Author

Henri Cohen

Subjects
Algebra, Kummer theory, Difference polynomials, Class field theory, Complex multiplication, Abelian group, Algebraic number field, Hypergeometric function, Kummer's function, Mathematics
Abstract

Class field theory deals with Abelian extensions of base fields. It gives complete answers to the existence of Abelian extensions with given relative or absolute discriminants. However, the algorithmic construction of these extensions is not completely straightforward. There are several ways to do this, but at present the most efficient general method is the use of Kummer extensions. In the next chapter, we will describe two other methods using analytic techniques, one using Stark units and Stark’s conjecture, the other using complex multiplication. Both of these methods impose restrictions on the base field, but when they are applicable they are much more efficient.

Published
2000

38. Appendix C: Tables

Author

Henri Cohen

Subjects
Discrete mathematics, medicine.anatomical_structure, medicine, Appendix, Mathematics
Published
2000

39. The Fundamental Theorems of Global Class Field Theory

Author

Henri Cohen

Subjects
Algebra, Fundamental theorem, Prime ideal, Class field theory, Galois group, Algebraic number field, Mathematical proof, Mathematics, Congruence subgroup
Abstract

In this chapter, we give the main results of global class field theory for the case of number fields. We refer the reader to [Art-Tat], [Gras], [Has1], [Jan], or [Mart4] for more detailed statements and proofs. We present the results “a la Hasse”, without using ideles. This is more suitable for algorithmic treatment. For an idelic treatment, we refer to [Neu]. I have largely benefited from the notes of J. Martinet [Mart4] in writing this chapter.

Published
2000

40. Variations on Class and Unit Groups

Author

Henri Cohen

Subjects
Combinatorics, Class (set theory), Modulo, Class field theory, Abelian group, Unit (ring theory), Mathematics
Abstract

In Chapter 3 we studied variants of class and unit groups, the ray class groups Clm(K), as well as the associated unit groups Um(K) of units multiplicatively congruent to 1 modulo m. The fundamental application of these notions through the deep theorems of class field theory is the construction of Abelian extensions.

Published
2000

41. Computational Class Field Theory

Author

Peter Stevenhagen and Henri Cohen

Subjects
Algebra, Class (set theory), Kummer theory, Discriminant, Non-abelian class field theory, Class field theory, Abelian group, Algebraic number field, Principal ideal theorem, Mathematics
Abstract

In Chapter 3 we gave the main theoretical results concerning global class field theory over number fields. We are now going to study this subject from the algorithmic point of view. In the present chapter, we give efficient algorithms for computing ray class groups of number fields and for computing the conductor and norm group of the Abelian extensions corresponding to congruence subgroups of these ray class groups by Takagi’s Theorem 3.5.1. Thanks to Proposition 3.5.8 and Theorem 3.5.11, this allows us to compute their signature and discriminant.

Published
2000

42. Convergence acceleration of alternating series

Author

Fernando Rodriguez Villegas, Henri Cohen, and Don Zagier

Subjects
Discrete mathematics, Class (set theory), Chebyshev polynomials, Convergence acceleration, General Mathematics, 11Y55, alternating sum, Chebyshev polynomial, Combinatorics, Alternating series, Alternating series test, Number theory, Linear acceleration, Mathematics, 65B05
Abstract

We discuss some linear acceleration methods for alternating series which are in theory and in practice much better than that of Euler--Van Wijngaarden. One of the algorithms, for instance, allows one to calculate $\sum(-1)^ka_k$ with an error of about $17$.$93^{-n}$ from the first $n$ terms for a wide class of sequences $\{a_k\}$. Such methods are useful for high precision calculations frequently appearing in number theory.

Published
2000

43. Appendix A: Theoretical Results

Author

Henri Cohen

Subjects
Pure mathematics, symbols.namesake, medicine.anatomical_structure, Prime ideal, Galois group, Normal extension, medicine, symbols, Algebraic number field, Appendix, Dirichlet series, Mathematics
Published
2000

44. Computing Defining Polynomials Using Analytic Methods

Author

Henri Cohen

Subjects
Pure mathematics, Kummer theory, Difference polynomials, Class field theory, Field (mathematics), Quadratic field, Abelian group, Algebraic number field, Mathematics, Congruence subgroup
Abstract

In the preceding chapter we saw how to construct Abelian extensions of a number field K corresponding to a given congruence subgroup (m, C) by using Kummer theory. The main advantage of this method is that it is completely general (and hence is the only method used for the proofs of the existence results in class field theory), but its main drawback is the necessity of working over a larger field K z = K(ζn).

Published
2000

45. Cubic Number Fields

Author

Henri Cohen

Subjects
Pure mathematics, Quadratic equation, Quadratic form, Quadratic field, Isomorphism class, Algebraic number field, Mathematics
Abstract

In [Coh0, Chapter 5], we studied quadratic fields in great detail. The goal of the present chapter is to do the same for cubic fields. We have already studied them in some detail in [Coh0, Chapter 6], but in the present chapter we will deal with deeper subjects and also show how to generate tables of cubic fields almost as efficiently as tables of quadratic fields. The spirit of this chapter is slightly different from that of the preceding chapters, which essentially deal with relative extensions, but the results are sufficiently important to be included in a textbook. Had they been known when [Coh0] was first published, they would, of course, have been included there.

Published
2000

46. Counting Discriminants of Number Fields of Degree up to Four

Author

Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier

Subjects
Combinatorics, Degree (graph theory), Galois cohomology, Mathematics::Number Theory, Galois group, Galois extension, Permutation group, Algebraic number field, Galois module, Resolvent, Mathematics
Abstract

For each permutation group G on n letters with n ≤ 4, we give results, conjectures and numerical computations on discriminants of number fields L of degree n over ℚ such that the Galois group of the Galois closure of L is isomorphic to G.

Published
2000

47. A table of totally complex number fields of small discriminants

Author

Michel Olivier, Francisco Diaz y Diaz, and Henri Cohen

Subjects
Pure mathematics, Discriminant, Degree (graph theory), Class field theory, Root (chord), Algebraic number field, Table (information), Complex number, Mathematics
Abstract

Using the explicit class field theory developed in [3] and tables of number fields in low degree, we construct totally complex number fields having a degree smaller than 80 and a root discriminant near from Odlyzko's bounds. For some degrees, we extend and improve the table of totally complex number fields of small discriminants given by Martinet

Published
1998

48. Efficient Elliptic Curve Exponentiation Using Mixed Coordinates

Author

Takatoshi Ono, Atsuko Miyaji, and Henri Cohen

Subjects
Discrete mathematics, Elliptic curve point multiplication, Jacobian curve, Curve25519, Elliptic Curve Digital Signature Algorithm, Hessian form of an elliptic curve, Hardware_ARITHMETICANDLOGICSTRUCTURES, Schoof's algorithm, Elliptic curve cryptography, Tripling-oriented Doche–Icart–Kohel curve, Computer Science::Cryptography and Security, Mathematics
Abstract

Elliptic curve cryptosystems, proposed by Koblitz ([12]) and Miller ([16]), can be constructed over a smaller field of definition than the ElGamal cryptosystems ([6]) or the RSA cryptosystems ([20]). This is why elliptic curve cryptosystems have begun to attract notice. In this paper, we investigate efficient elliptic curve exponentiation. We propose a new coordinate system and a new mixed coordinates strategy, which significantly improves on the number of basic operations needed for elliptic curve exponentiation.

Published
1998

49. Computing in Algebraic Number Fields

Author

Henri Cohen

Subjects
Algebra, Function field of an algebraic variety, Algebraic operation, Real algebraic geometry, Algebraic extension, Dimension of an algebraic variety, Field (mathematics), Algebraic number, Mathematics
Published
1998

50. Imprimitive octic fields with small discriminants

Author

Michel Olivier, Francisco Diaz y Diaz, and Henri Cohen

Subjects
Pure mathematics, Mathematics::Algebraic Geometry, Mathematics::Number Theory, Mirror effect, Quartic function, Class field theory, Galois group, Transitive group, Class number, Dedekind zeta function, Mathematics
Abstract

We give here the complete table of octic fields with a quartic subfield with minimum discriminants for all signatures and for all the possible Galois groups. Moreover we give some pairs of octic fields arithmetically equivalent (i.e. with the same Dedekind zeta function).

Published
1998

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