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1. Arithmetic Geometry: Computation and Applications

Author

Yves Aubry, Christophe Ritzenthaler, Everett W. Howe, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Computer science, Computation, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Arithmetic, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract

International audience

Published
2019

2. Principally polarized squares of elliptic curves with field of moduli equal to Q

Author

Everett W. Howe, Alexandre Gélin, Christophe Ritzenthaler, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Field (physics), Mathematics - Number Theory, 010102 general mathematics, Mathematical analysis, 010103 numerical & computational mathematics, 01 natural sciences, 11G15, 14H25, 14H45, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Moduli, Elliptic curve, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, FOS: Mathematics, Number Theory (math.NT), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Mathematics
Abstract

Accepted for the Proceedings of ANTS-13; International audience; We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the Generalized Riemann Hypothesis is true, there are no further examples of such curves. More generally, we prove under the Generalized Riemann Hypothesis that there exist exactly 46 genus-2 curves over $\overline{\mathbb{Q}}$ with field of moduli $\mathbb{Q}$ whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.

Published
2018

3. Algebraic Geometry for Coding Theory and Cryptography : IPAM, Los Angeles, CA, February 2016

Author

Everett W. Howe, Kristin E. Lauter, Judy L. Walker, Everett W. Howe, Kristin E. Lauter, and Judy L. Walker

Subjects
Cryptography--Congresses, Geometry, Algebraic--Congresses, Coding theory--Congresses
Abstract

Covering topics in algebraic geometry, coding theory, and cryptography, this volume presents interdisciplinary group research completed for the February 2016 conference at the Institute for Pure and Applied Mathematics (IPAM) in cooperation with the Association for Women in Mathematics (AWM). The conference gathered research communities across disciplines to share ideas and problems in their fields and formed small research groups made up of graduate students, postdoctoral researchers, junior faculty, and group leaders who designed and led the projects. Peer reviewed and revised, each of this volume's five papers achieves the conference's goal of using algebraic geometry to address a problem in either coding theory or cryptography. Proposed variants of the McEliece cryptosystem based on different constructions of codes, constructions of locally recoverable codes from algebraic curves and surfaces, and algebraic approaches to the multicast network coding problem are only some of the topics covered in this volume. Researchers and graduate-level students interested in the interactions between algebraic geometry and both coding theory and cryptography will find this volume valuable.

Published
2017

4. Split abelian surfaces over finite fields and reductions of genus-2 curves

Author

Jeffrey D. Achter and Everett W. Howe

Subjects
11G20, reduction, simplicity, 01 natural sciences, Prime (order theory), Combinatorics, symbols.namesake, Mathematics - Algebraic Geometry, 14K15, 11G10, 11G20, 11G30, Genus (mathematics), 0103 physical sciences, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Abelian group, Algebraic Geometry (math.AG), curve, Mathematics, counting function, Algebra and Number Theory, Conjecture, Mathematics - Number Theory, 11G30, abelian surface, 11G10, 010102 general mathematics, Function (mathematics), Algebraic number field, reducibility, Riemann hypothesis, Finite field, symbols, 010307 mathematical physics, 14K15, Jacobian
Abstract

For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field F_q is not simple. We show that there are positive constants B and C such that for all q, B (log q)^{-3}(log log q)^{-4} < s(q)sqrt(q) < C (log q)^4(log log q)^2, and we obtain better estimates under the assumption of the generalized Riemann hypothesis. If A is a principally-polarized abelian surface over a number field K, let pi_split(A/K, z) denote the number of prime ideals p of K of norm at most z such that A has good reduction at p and A_p is not simple. We conjecture that for sufficiently general A, the counting function pi_split(A/K, z) grows like sqrt(z)/log z. We indicate why our theorem on the rate of growth of s(q) gives us reason to hope that our conjecture is true., 27 pages, 3 figures. V2 incorporates new data courtesy of Andrew Sutherland

Published
2017

5. Genus-2 Jacobians with torsion points of large order

Author

Everett W. Howe

Subjects
Pure mathematics, Rational number, General Mathematics, 14G05 (Primary) 11G30, 14H25, 14H45 (Secondary), Elliptic curve, symbols.namesake, Mathematics - Algebraic Geometry, Rational point, Jacobian matrix and determinant, Torsion (algebra), symbols, FOS: Mathematics, Algebraic Geometry (math.AG), Mathematics
Abstract

We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational point of order 48, parametrized by a rank-2 elliptic curve over Q, and we exhibit a single genus-2 curve over Q whose Jacobian has a rational point of order 70, the largest order known. We also give new examples of genus-2 Jacobians with rational points of order 27, 28, and 39. Most of our examples are produced by `gluing' two elliptic curves together along their n-torsion subgroups, where n is either 2 or 3. The 2-gluing examples arise from techniques developed by the author in joint work with Lepr\'evost and Poonen 15 years ago. The 3-gluing examples are made possible by an algorithm for explicit 3-gluing over non-algebraically closed fields recently developed by the author in joint work with Br\"oker, Lauter, and Stevenhagen., Comment: Updated references to include a paper of Platonov, Zhgun, and Petrunin that gives two curves we had thought had not been found before

Published
2014

6. Genus bounds for curves with fixed Frobenius eigenvalues

Author

Christophe Ritzenthaler, Everett W. Howe, Noam D. Elkies, Institut de mathématiques de Luminy (IML), Centre National de la Recherche Scientifique (CNRS)-Université de la Méditerranée - Aix-Marseille 2, and Université de la Méditerranée - Aix-Marseille 2-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Linear programming, General Mathematics, Mathematics::Number Theory, 0102 computer and information sciences, 01 natural sciences, Upper and lower bounds, Finite collection, symbols.namesake, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Genus (mathematics), FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Abelian group, [MATH]Mathematics [math], Mathematics::Representation Theory, Algebraic Geometry (math.AG), Eigenvalues and eigenvectors, ComputingMilieux_MISCELLANEOUS, Mathematics, Mathematics - Number Theory, Applied Mathematics, 010102 general mathematics, 14G10, 11G20, 14G15, 14H25, 010201 computation theory & mathematics, Product (mathematics), Jacobian matrix and determinant, symbols
Abstract

For every finite collection C of abelian varieties over F_q, we produce an explicit upper bound on the genus of curves over F_q whose Jacobians are isogenous to a product of powers of elements of C., Comment: 14 pages. The new version includes new references to papers with related results; also, the main result has been greatly improved, thanks to an argument suggested by Zeev Rudnick and Sergei Konyagin

Published
2014

7. Jacobians in isogeny classes of abelian surfaces over finite fields

Author

Enric Nart, Everett W. Howe, Christophe Ritzenthaler, Institut de mathématiques de Luminy (IML), Université de la Méditerranée - Aix-Marseille 2-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de la Méditerranée - Aix-Marseille 2, and Ritzenthaler, Christophe

Subjects
Isogeny, 11G20 (Primary) 14G10, 14G15 (Secondary), Pure mathematics, Algebra and Number Theory, Mathematics - Number Theory, 010102 general mathematics, [MATH] Mathematics [math], 0102 computer and information sciences, 01 natural sciences, Algebra, Mathematics - Algebraic Geometry, Finite field, 010201 computation theory & mathematics, FOS: Mathematics, Geometry and Topology, Number Theory (math.NT), [MATH]Mathematics [math], 0101 mathematics, Abelian group, Algebraic Geometry (math.AG), ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields., LaTeX, 38 pages. Intermediate results in Section 13 have been generalized. Minor changes elsewhere

Published
2006

8. Curves of every genus with many points. II. Asymptotically good families

Author

Everett W. Howe, Noam D. Elkies, Michael E. Zieve, Joseph L. Wetherell, Bjorn Poonen, Andrew Kresch, and University of Zurich

Subjects
2 covering of curves, 11G20, General Mathematics, 14G15, 0102 computer and information sciences, 14G05 (Primary) 11G20, 14G15 (Secondary), 01 natural sciences, Combinatorics, Mathematics - Algebraic Geometry, symbols.namesake, 510 Mathematics, Integer, Genus (mathematics), FOS: Mathematics, asymptotic lower bounds, Number Theory (math.NT), 0101 mathematics, Algebraic Geometry (math.AG), Prime power, Mathematics, 2600 General Mathematics, Mathematics - Number Theory, 010102 general mathematics, degree, 10123 Institute of Mathematics, Finite field, 010201 computation theory & mathematics, Jacobian matrix and determinant, class field towers, symbols, Constant (mathematics), curves over finite fields with many rational points, 14G05
Abstract

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = d * (log q). We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over F_q that has at least c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > c*g/n., LaTeX, 18 pages

Published
2004

9. Lower bounds on the lengths of double-base representations.

Author

Vassil S. Dimitrov and Everett W. Howe

Subjects
*REPRESENTATIONS of algebras, *PRIME numbers, *NUMBER theory, *MATHEMATICAL analysis, *NUMERICAL analysis, *ALGEBRA
Abstract

A double-base representation of an integer $ n$ $ n = n_1 + \cdots + n_r$ are (positive or negative) integers that are divisible by no primes other than $ 2$; the length of the representation is the number $ r$ such that every integer $ n$ $ a\log n / \log\log n$ such that there are infinitely many integers $ n$ $ c\log n / (\log\log n \log\log\log n)$ Our methods allow us to find the smallest positive integers with no double-base representations of several lengths. In particular, we show that $ 103$, that $ 4985$, that $ 641687$, and that $ 326552783$. [ABSTRACT FROM AUTHOR]

Published
2011
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10. Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian.

Author

Everett W. Howe

Subjects
JACOBIAN matrices, QUARTIC equations
Abstract

We show how for every integer $n$ one can explicitly construct $n$ distinct plane quartics and one hyperelliptic curve over ${\mathbf C}$ all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly'', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in ${\mathbf R}$ whose minimal polynomials over ${\mathbf Q}$ can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over ${\mathbf C}$, each with a pair of commuting involutions, are isomorphic to one another. [ABSTRACT FROM AUTHOR]

Published
2001
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12. On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

Author

Hui June Zhu and Everett W. Howe

Subjects
Discrete mathematics, Algebra and Number Theory, Torsion subgroup, Mathematics - Number Theory, Elementary abelian group, Rank of an abelian group, Free abelian group, Combinatorics, Mathematics - Algebraic Geometry, Abelian variety, 14G15 (Primary) 11G10, 14K15 (Secondary), Abelian variety of CM-type, FOS: Mathematics, Genus field, absolute simplicity, Number Theory (math.NT), Abelian group, Algebraic Geometry (math.AG), finite field, Arithmetic of abelian varieties, Mathematics
Abstract

We prove that for every field k and every positive integer n, there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k,n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n, the quantity S(F_q,n) approaches 1 as q approaches infinity over the prime powers., 17 pages, AMS-LaTeX

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13. Nonisomorphic curves that become isomorphic over extensions of coprime degrees

Author

Everett W. Howe, Michael E. Zieve, Robert M. Guralnick, and Daniel Goldstein

Subjects
Discrete mathematics, Algebra and Number Theory, Coprime integers, Degree (graph theory), Galois cohomology, Field (mathematics), Automorphism, 14H37 (Primary) 14H25, 14H45 (Secondary), Mathematics - Algebraic Geometry, Finite field, Curve, Genus (mathematics), FOS: Mathematics, Twist, Algebraic Geometry (math.AG), Mathematics
Abstract

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K_0 be an arbitrary prime field and let r and s be integers greater than 1 that are coprime to one another. We show that one can find a finite extension K of K_0, a degree-r extension L of K, a degree-s extension M of K, and two curves C and D over K such that C and D become isomorphic to one another over L and over M, but not over any proper subextensions of L/K or M/K. We show that such C and D can never have genus 0, and that if K is finite, C and D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degree extension of F_3. On the other hand, when {r,s}={2,3} we show that genus-2 examples occur in every characteristic other than 3. Our detailed analysis of the case {r,s} = {2,3} shows that over every finite field K there exist nonisomorphic curves C and D that become isomorphic to one another over the quadratic and cubic extensions of K. Most of our proofs rely on Galois cohomology. Without using Galois cohomology, we show that two nonisomorphic genus-0 curves over an arbitrary field remain nonisomorphic over every odd-degree extension of the base field., LaTeX, 32 pages. Further references added to the discussion in Section 10

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14. Constructing Distinct Curves with Isomorphic Jacobians

Author

Everett W. Howe

Subjects
Discrete mathematics, Pure mathematics, symbols.namesake, Algebra and Number Theory, Mathematics::Algebraic Geometry, Simple (abstract algebra), Plane (geometry), Quartic function, Genus (mathematics), Jacobian matrix and determinant, symbols, Field (mathematics), Mathematics
Abstract

We show that the hyperelliptic curvesy2=x5+x3+x2−x−1 andy2=x5−x3+x2−x−1 over the field with three elements are not geometrically isomorphic, and yet they have isomorphic Jacobian varieties. Furthermore, their Jacobians are absolutely simple. We present a method for constructing further such examples. We also present two curves of genus three, one hyperelliptic and one a plane quartic, that have isomorphic absolutely simple Jacobians

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15. 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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16. Improved CRT Algorithm for Class Polynomials in Genus $2$

Author

Damien Robert, Kristin E. Lauter, Everett W. Howe, Kiran S. Kedlaya, Cryptography group, Microsoft Research [Redmond], Microsoft Corporation [Redmond, Wash.]-Microsoft Corporation [Redmond, Wash.], 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), ANR-12-BS01-0010,PEACE,Espaces de paramètres pour une arithmétique efficace et une évaluation de la sécurité des courbes(2012), European Project: 278537,EC:FP7:ERC,ERC-2011-StG_20101014,ANTICS(2012), 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
Discrete mathematics, Isogeny, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Class (set theory), Group (mathematics), Generalization, Mathematics::Number Theory, 010102 general mathematics, Order (ring theory), 01 natural sciences, Randomized algorithm, 010101 applied mathematics, Class polynomials, Mathematics::Algebraic Geometry, Genus (mathematics), 0101 mathematics, Endomorphism ring, Algorithm, Mathematics
Abstract

International audience; We present a generalization to genus~2 of the probabilistic algorithm of Sutherland for computing Hilbert class polynomials. The improvement over the Br{ö}ker-Gruenewald-Lauter algorithm for the genus~2 case is that we do not need to find a curve in the isogeny class whose endomorphism ring is the maximal order; rather, we present a probabilistic algorithm for ''going up'' to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of $(\ell,\ell)$-isogenies to compute all isogenous maximal curves from an initial one. This is an extended version of the article published at ANTS~X.

Published
2013

17. Computing equations of curves with many points

Author

Ducet, Virgile, Fieker, Claus, Institut de mathématiques de Luminy (IML), Centre National de la Recherche Scientifique (CNRS)-Université de la Méditerranée - Aix-Marseille 2, Technische Universität Kaiserslautern (TU Kaiserslautern), Everett W. Howe and Kiran S. Kedlaya, and Université de la Méditerranée - Aix-Marseille 2-Centre National de la Recherche Scientifique (CNRS)

Subjects
11R37, 14G05, 14H30, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract

International audience; We explain how to compute the equations of the abelian coverings of any curve defined over a finite field. Then we describe an algorithm which computes curves with many rational points with respect to their genus. The implementation of the algorithm provides 7 new records over F 2 .

Published
2013

18. Imaginary quadratic fields with isomorphic abelian Galois groups

Author

Athanasios Angelakis, Peter Stevenhagen, 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), Mathematical institute, Universiteit Leiden [Leiden], Everett W. Howe and Kiran S. Kedlaya, 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, 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 Universiteit Leiden

Subjects
Pure mathematics, Profinite group, Mathematics - Number Theory, group extensions, Mathematics::Number Theory, 010102 general mathematics, Galois group, 010103 numerical & computational mathematics, Absolute Galois group, Basis (universal algebra), Algebraic number field, 16. Peace & justice, 01 natural sciences, 11R37 (Primary) 20K35 (Secondary), [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 2000 Mathematics Subject Classification. Primary 11R37, Secondary 20K35, class field theory, Character (mathematics), absolute Galois group, Class field theory, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Abelian group, Mathematics
Abstract

In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In this paper, we provide a direct `computation' of the profinite group $A_K$ for imaginary quadratic $K$, and use it to obtain many different $K$ that all have the same minimal absolute abelian Galois group., Comment: 17 pages; to appear in the proceedings volume of ANTS-X, San Diego 2012

Published
2012

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