Descriptor: "[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]" - Searchworks@Jio Institute Digital Library Search Results

Your search keyword '"[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]"' showing total 201 results

Start Over Descriptor "[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]"

201 results on '"[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]"'

1. A bound on the primes of bad reduction for CM curves of genus 3

Author: Kristin E. Lauter, Rachel Newton, Marco Streng, Pınar Kılıçer, Ekin Ozman, Elisa Lorenzo García, Carl Von Ossietzky Universität Oldenburg, Cryptography group, Microsoft Research [Redmond], Microsoft Corporation [Redmond, Wash.]-Microsoft Corporation [Redmond, Wash.], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics and Statistics, University of Reading, RG66AX, Reading UK, Bogazici University [Istanbul], Universiteit Leiden [Leiden], Microsoft Corporation [Redmond], Microsoft Corporation [Redmond, Wash.], Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), University of Reading (UOR), Boğaziçi University [Istanbul], 639.031.243, Engineering and Physical Sciences Research Council, Carl Von Ossietzky Universität Oldenburg = Carl von Ossietzky University of Oldenburg (OFFIS), 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), Boǧaziçi üniversitesi = Boğaziçi University [Istanbul], Universiteit Leiden, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), and Algebra
Subjects: Reduction (recursion theory), General Mathematics, Mathematics::Number Theory, 010103 numerical & computational mathematics, Algebraic geometry, 01 natural sciences, Prime (order theory), Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Genus (mathematics), FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Abelian group, Algebraic Geometry (math.AG), Mathematics, Mathematics - Number Theory, Applied Mathematics, 010102 general mathematics, 11G10, 11G15, 14H45, 14K22, 14J15, 14Q05, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Elliptic curve, Finite field, Number theory, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields., Comment: 16 pages, some minor updates, some typos are corrected
Published: 2020
Full Text: View/download PDF

2. Primes Dividing Invariants of CM Picard Curves

Author: Pınar Kılıçer, Elisa Lorenzo García, Marco Streng, Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Universiteit Leiden [Leiden], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 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 ), Algebra, 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), Universiteit Leiden, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: General Mathematics, 010103 numerical & computational mathematics, Good reduction, Type (model theory), 01 natural sciences, 14G50, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, Reduction (complexity), Mathematics - Algebraic Geometry, math.AG, Mathematics::Algebraic Geometry, 14Q05, 14H45, 14K22, 11H06, 14G50, 14H40, 14Q05, Genus (mathematics), FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Algebraic Geometry (math.AG), Mathematics, 14H40, Mathematics - Number Theory, 14H45, 14K22, 010102 general mathematics, Complex multiplication, Mathematics::Geometric Topology, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], math.NT, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 11H06
Abstract: We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.
Published: 2019
Full Text: View/download PDF

3. Λ-ADIC FAMILIES OF JACOBI FORMS

Author: Longo, Matteo, Nicole, Marc-Hubert, Dipartimento di Matematica [padova], Universita degli Studi di Padova, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille ( I2M ), Aix Marseille Université ( AMU ) -Ecole Centrale de Marseille ( ECM ) -Centre National de la Recherche Scientifique ( CNRS ), Università degli Studi di Padova = University of Padua (Unipd), International Research Lab (IRL CRM-CNRS), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM)-Université de Montréal (UdeM)-Centre National de la Recherche Scientifique (CNRS), and International Research Laboratory (IRL) du CRM de Montréal, 3457, CRM-CNRS, Université de Montréal, Chemin de la Tour, Montréal
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics::Number Theory, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics::Representation Theory, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: International audience; We show that Hida's families of p-adic elliptic modular forms generalize to p-adic families of Jacobi forms. We also construct p-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.
Published: 2020
Full Text: View/download PDF

4. Choosing and generating parameters for pairing implementation on BN curves

Author: Sylvain Duquesne, Safia Haloui, Franck Rondepierre, Nadia El Mrabet, 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), Laboratoire International de Recherche en Informatique et Mathématiques Appliquées (LIRIMA), Université de Yaoundé I-Université Badji Mokhtar Annaba (UBMA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Joseph Ki-Zerbo [Ouagadougou] (UJZK)-Université d'Antananarivo-Université Gaston Bergé Sénégal-Centre National de la Recherche Scientifique et Technologique (CNRST), Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria), Département Systèmes et Architectures Sécurisés (SAS-ENSMSE), École des Mines de Saint-Étienne (Mines Saint-Étienne MSE), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-CMP-GC, Laboratoire d'Informatique Avancée de Saint-Denis (LIASD), Université Paris 8 Vincennes-Saint-Denis (UP8), Institut Mines-Télécom [Paris] (IMT), Danmarks Tekniske Universitet = Technical University of Denmark (DTU), Oberthur Technologies, ANR-12-INSE-0014,SIMPATIC,SIM et théorie des couplages pour la sécurité de l'information et des communications(2012), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Centre National de la Recherche Scientifique et Technologique (CNRST)-Université Gaston Bergé Sénégal-Université d'Antananarivo-Université Joseph Ki-Zerbo [Ouagadougou] (UJZK)-Université Badji Mokhtar - Annaba [Annaba] (UBMA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Yaoundé I, Danmarks Tekniske Universitet (DTU), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Université de Yaoundé I [Yaoundé]-Université Badji Mokhtar - Annaba [Annaba] (UBMA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Ouagadougou-Université d'Antananarivo-Université Gaston Bergé Sénégal-Centre National de la Recherche Scientifique et Technologique (CNRST), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire International de Recherche en Informatique et Mathématiques Appliquées ( LIRIMA ), Université de Yaoundé I [Yaoundé]-Université Badji Mokhtar - Annaba [Annaba] ( UBMA ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Ouagadougou-Université d'Antananarivo-Université Gaston Bergé Sénégal-Centre National pour la recherche scientifique et Technique, Département Systèmes et Architectures Sécurisés ( SAS-ENSMSE ), École des Mines de Saint-Étienne ( Mines Saint-Étienne MSE ), Institut Mines-Télécom [Paris]-Institut Mines-Télécom [Paris]-CMP-GC, Laboratoire d'Informatique Avancée de Saint-Denis ( LIASD ), Université Paris 8 Vincennes-Saint-Denis ( UP8 ), Danmarks Tekniske Universitet ( DTU ), and ANR-12-INSE-0014,SIMPATIC,SIM et théorie des couplages pour la sécurité de l'information et des communications ( 2012 )
Subjects: Elementary arithmetic, [ INFO.INFO-CR ] Computer Science [cs]/Cryptography and Security [cs.CR], Context (language use), Field (mathematics), [ MATH.MATH-IT ] Mathematics [math]/Information Theory [math.IT], 02 engineering and technology, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], pairing, Combinatorics, [INFO.INFO-CR]Computer Science [cs]/Cryptography and Security [cs.CR], elliptic curves, 0202 electrical engineering, electronic engineering, information engineering, ComputingMilieux_MISCELLANEOUS, 11T71, 14G50, Mathematics, Discrete mathematics, Algebra and Number Theory, Applied Mathematics, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], 020206 networking & telecommunications, Tower (mathematics), [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Elliptic curve, Finite field, Discrete logarithm, Pairing, Cryptography, 020201 artificial intelligence & image processing, finite fields
Abstract: Because pairings have many applications, many hardware and software pairing implementations can be found in the literature. However, the parameters generally used have been invalidated by the recent results on the discrete logarithm problem over pairing friendly elliptic curves (Kim and Barbulescu in CRYPTO 2016, volume 9814 of lecture notes in computer science, Springer, Berlin, pp 543–571, 2016). New parameters must be generated to insure enough security in pairing based protocols. More generally it could be useful to generate nice pairing parameters in many real-world applications (specific security level, resistance to specific attacks on a protocol, database of curves). The main purpose of this paper is to describe explicitly and exhaustively what should be done to generate the best possible parameters and to make the best choices depending on the implementation context (in terms of pairing algorithm, ways to build the tower field, $$\mathbb {F}_{p^{12}}$$ arithmetic, groups involved and their generators, system of coordinates). We focus on low level implementations, assuming that $$\mathbb {F}_p$$ additions have a significant cost compared to other $$\mathbb {F}_p$$ operations. However, our results are still valid if $$\mathbb {F}_p$$ additions can be neglected. We also explain why the best choice for the polynomials defining the tower field $$\mathbb {F}_{p^{12}}$$ is only dependent on the value of the BN parameter u mod small integers (like 12 for instance) as a nice application of old elementary arithmetic results. This should allow a faster generation of this parameter. Moreover, we use this opportunity to give some new slight improvements on $$\mathbb {F}_{p^{12}}$$ arithmetic (in a pairing context).
Published: 2017
Full Text: View/download PDF

5. The p-adic Variation of the Gross-Kohnen-Zagier Theorem

Author: Longo, Matteo, Nicole, Marc-Hubert, Dipartimento di Matematica [padova], Università degli Studi di Padova = University of Padua (Unipd), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Universita degli Studi di Padova, Institut de Mathématiques de Marseille ( I2M ), and Aix Marseille Université ( AMU ) -Ecole Centrale de Marseille ( ECM ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Number Theory, Mathematics::Number Theory, FOS: Mathematics, Number Theory (math.NT), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics::Representation Theory, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: We relate $p$-adic families of Jacobi forms to Big Heegner points constructed by B. Howard, in the spirit of the Gross-Kohnen-Zagier theorem. We view this as a GL(2) instance of a $p$-adic Kudla program., 16 pages
Published: 2019
Full Text: View/download PDF

6. Functorial properties of generalised Steinberg representations

Author: Tobias Schmidt, Claus Sorensen, Julien Hauseux, King‘s College London, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), University of California [San Diego] (UC San Diego), University of California-University of California, EP/L025302/1, Engineering and Physical Sciences Research Council, 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), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)
Subjects: Reductive groups, 010103 numerical & computational mathematics, Commutative ring, 01 natural sciences, Extensions, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], FOS: Mathematics, Number Theory (math.NT), Representation Theory (math.RT), 0101 mathematics, Steinberg representation, generalised Steinberg representations, Mathematics, Deformations, 22E50, 11F70, Algebra and Number Theory, Functor, Mathematics - Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, Reductive group, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Local fields, [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT], 22E50, Generalised Steinberg representations, Mathematics - Representation Theory
Abstract: Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$ containing $P$. We study the functor $\mathrm{St}_Q^G$ taking a smooth $R$-representation $\sigma$ of $L$ which extends to a representation $\mathrm{e}_G(\sigma)$ of $G$ trivial on $U$ to the smooth $R$-representation $\mathrm{e}_G(\sigma) \otimes_R \mathrm{St}_Q^G(R)$ of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation., Comment: 14 pages, minor changes, final version
Published: 2019
Full Text: View/download PDF

7. Iwasawa Theory and $F$-Analytic Lubin-Tate $(\varphi,\Gamma)$-Modules

Author: Berger , Laurent, Fourquaux , Lionel, Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-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), ANR-14-CE25-0002,PerCoLaTor,PERfectoïdes, cohomologie COmplétée, correspondance de LAnglands et cohomologie de TORsion(2014), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut Universitaire de France ( IUF ), Ministère de l'Éducation nationale, de l’Enseignement supérieur et de la Recherche ( M.E.N.E.S.R. ), Unité de Mathématiques Pures et Appliquées ( UMPA-ENSL ), École normale supérieure - Lyon ( ENS Lyon ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-14-CE25,PerCoLaTor,PERfectoides COrresondance de LAnglands et TORsion dans la cohomologie ( 2014 ), Guillemer, Marie-Annick, and Appel à projets générique - PERfectoïdes, cohomologie COmplétée, correspondance de LAnglands et cohomologie de TORsion - - PerCoLaTor2014 - ANR-14-CE25-0002 - Appel à projets générique - VALID
Subjects: Mathematics - Number Theory, analytic cohomology, Mathematics::Number Theory, General Mathematics, 11FXX, 11SXX, 14GXX, Lubin-Tate group, Mathematics::Algebraic Topology, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Bloch-Kato exponential, Mathematics::K-Theory and Homology, overconvergent representation, normalized traces, p-adic Hodge theory, Kummer theory, p-adic representation, 11F, 11S, 14G, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], Iwasawa theory
Abstract: Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(\varphi,\Gamma)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting., Comment: v2: final version, to appear in Documenta Mathematica. 31 pages
Published: 2017
Full Text: View/download PDF

8. The Zeta Mahler measure of $(z^n − 1)/(z − 1)$

Author: Arunabha Biswas and M. Ram Murty
Subjects: [ MATH ] Mathematics [math], Sequence, Polynomial, Conjecture, Mathematics::Number Theory, Laurent polynomial, Generating function, higher Mahler measure 2010 Mathematics Subject Classification 11R06, Mathematics::Geometric Topology, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, Unit circle, Mahler measure, Computer Science::Symbolic Computation, 11M99, Mathematics
Abstract: We consider the k-higher Mahler measure $m_k (P) $ of a Laurent polynomial $P$ as the integral of ${\log}^k |P | $ over the complex unit circle and zeta Mahler measure as the generating function of the sequence ${m_k (P)}$. In this paper we derive a few properties of the zeta Mahler measure of the polynomial $P_n (z) := (z^N − 1)/(z − 1) $ and propose a conjecture.
Published: 2019
Full Text: View/download PDF

9. Set Equidistribution of subsets of (Z/nZ) *

Author: Veekesh Kumar, Jaitra Chattopadhyay, Ravindranathan Thangadurai, and Harish-Chandra Research Institute
Subjects: [ MATH ] Mathematics [math], Class (set theory), Modulo, Mathematics::Number Theory, Cyclic group, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Combinatorics, Set (abstract data type), [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Order (group theory), Coset, [MATH]Mathematics [math], Mathematics, Complement (set theory)
Abstract: In 2010, Murty and Thangadurai [MuTh10] provided a criterion for the set equidistribution of residue classes of subgroups in (Z/nZ) *. In this article, using similar methods, we study set equidistribution for some class of subsets of (Z/nZ) *. In particular, we study the set equidistribution modulo 1 of cosets, complement of subgroups of the cyclic group (Z/nZ) * and the subset of elements of fixed order, whenever the size of the subset is sufficiently large.
Published: 2019

10. A note on some congruences involving arithmetic functions

Author: József Sándor
Subjects: [ MATH ] Mathematics [math], Divisor function, 11A07, Euler's totient function, Function (mathematics), Congruence relation, 11D45, Euler's totient, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, congruences 2010 Mathematics Subject Classification 11A25, 11N05, primality, symbols, Congruence (manifolds), Arithmetic function, Dual polyhedron, Dedekind cut, Dedekind's arithmetical function, number of divisors, Mathematics
Abstract: We consider some congruences involving arithmetical functions. For example, we study the congruences nψ(n) ≡ 2 (mod ϕ(n)), nϕ(n) ≡ 2 (mod ψ(n)), ψ(n)d(n) − 2 ≡ 0 (mod n), where ϕ(n), ψ(n), d(n) denote Euler's totient, Dedekind's function, and the number of divisors of n, respectively. Two duals of the Lehmer congruence n − 1 ≡ 0 (mod ϕ(n)) are also considered.
Published: 2019
Full Text: View/download PDF

11. On the Wintner-Ingham-Segal summability method

Author: Yoshio Tanigawa, Shigeru Kanemitsu, T Kuzumaki, Kyushu Institute of Technology, Faculty of Engineering, Gifu University, Graduate School of Mathematics [Nagoya], and Nagoya University
Subjects: [ MATH ] Mathematics [math], Pure mathematics, Stieltjes integral, Mathematics::Classical Analysis and ODEs, Riemann–Stieltjes integral, (Ingham) summability, Term (logic), [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics::K-Theory and Homology, Subject (grammar), weak Riemann hypothesis 2010 Mathematics Subject Classification 40D99, [MATH]Mathematics [math], 11M99, zero-free region, Mathematics, Prime number theorem
Abstract: The aim of this note is to establish a subclass of $\mathcal{F}$ considered by Segal if functions for which the Ingham-Wintner summability implies $\mathcal{F}$-summability as wide as possible. The subclass is subject to the estimate for the error term of the prime number theorem. We shall make good use of Stieltjes integration which elucidates previous results obtained by Segal.
Published: 2019
Full Text: View/download PDF

12. On-Regular Bipartitions Modulo $m$

Author: K N Vidya and D D Somashekara
Subjects: [ MATH ] Mathematics [math], Modulo, regular partition, Congruence relation, regular bipartition, partition, Combinatorics, Congruence, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], 05A17, theta functions 2010 Mathematics Subject Classification 11P83, Partition (number theory), Congruence (manifolds), Mathematics
Abstract: Let $b_l (n)$ denote the number of $l$-regular partitions of $n$ and $B_l (n)$ denote the number of $l$-regular bipartitions of $n$. In this paper, we establish several infinite families of congruences satisfied by $B_l (n)$ for $l \in {2, 4, 7}$. We also establish a relation between $b_9 (2n)$ and $B_3 (n)$.
Published: 2019
Full Text: View/download PDF

13. Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions

Author: Yeongseong Jo, Kevin Nowland, and Keshav Aggarwal
Subjects: [ MATH ] Mathematics [math], Coprime integers, Mathematics::Number Theory, 010102 general mathematics, Prime number, Holomorphic function, 11F67, Square-free integer, 01 natural sciences, Cusp form, Dirichlet character, Combinatorics, 11L05, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Special values of L-functions, 0103 physical sciences, δ-method 2010 Mathematics Subject Classification 11F11, Second moment method, 010307 mathematical physics, 0101 mathematics, subconvexity, Mathematics, Rankin-Selberg convolution
Abstract: Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P \sim M^{\eta}$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1]{2}, f \otimes \chi)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $\chi$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.
Published: 2019
Full Text: View/download PDF

14. Integral points on circles

Author: Mariusz Skałba, Andrzej Schinzel, Institut of Mathematics - Polish Academy of Sciences (PAN), Polska Akademia Nauk = Polish Academy of Sciences (PAN), and University of Warsaw (UW)
Subjects: [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, 11D09, Fermat's theorem on sums of two squares, Gaussian integers 2010 Mathematics Subject Classification 11D25, Square-free integer, Radius, [MATH]Mathematics [math], sums of two squares, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Mathematics
Abstract: Sixty years ago the first named author gave an example \cite{sch} of a circle passing through an arbitrary number of integral points. Now we shall prove: {\it The number $N$ of integral points on the circle $(x-a)^2+(y-b)^2=r^2$ with radius $r=\frac{1}{n}\sqrt{m}$, where $m,n\in\mathbb Z$, $m,n>0$, $\gcd(m,n^2)$ squarefree and $a,b\in\mathbb Q$ does not exceed $r(m)/4$, where $r(m)$ is the number of representations of $m$ as the sum of two squares, unless $n|2$ and $n\cdot (a,b)\in\mathbb Z^2$; then $ N\leq r(m)$}.
Published: 2019

15. Two applications of number theory to discrete tomography

Author: Robert Tijdeman
Subjects: Structure (mathematical logic), [ MATH ] Mathematics [math], 15A06, business.industry, Discrete tomography, Too quickly, Object (computer science), switching components 2010 Mathematics Subject Classification 94A08, sums of two squares, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Number theory, Computer vision, Artificial intelligence, Tomography, business, Mathematics
Abstract: Tomography is the theory behind scans, e.g. MRI-scans. Most common is continuous tomography where an object is reconstructed from numerous projections. In some cases this is not applicable, because the object changes too quickly or is damaged by making hundreds of projections (by X-rays). In such cases discrete tomography may apply where only few projections are made. The present paper shows how number theory helps to provide insight in the application and structure of discrete tomography.
Published: 2019
Full Text: View/download PDF

16. A Theorem of Fermat on Congruent Number Curves

Author: Lorenz Halbeisen and Norbert Hungerbühler
Subjects: Congruent numbers, [ MATH ] Mathematics [math], Fermat's Last Theorem, 2010 Mathematics Subject Classification. primary 11G05, secondary 11D25, Mathematics - Number Theory, Mathematics::General Mathematics, Mathematics::Number Theory, 010102 general mathematics, Order (ring theory), 010103 numerical & computational mathematics, Pythagorean triple, 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, 11G05, 11D25, Integer, Rational point, Elementary proof, FOS: Mathematics, Mathematics::Metric Geometry, Number Theory (math.NT), 0101 mathematics, Mathematics, Congruent number
Abstract: A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order., Comment: 8 pages, one figure
Published: 2019
Full Text: View/download PDF

17. On Pillai's problem with Pell numbers and powers of 2

Author: Alain Togbé, Florian Luca, Mohand Ouamar Hernane, and Salah Eddine Rihane
Subjects: 010101 applied mathematics, Discrete mathematics, reduction method 2010 Mathematics Subject Classification primary 11B39, secondary 11J86, [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], 010102 general mathematics, 0101 mathematics, Pell numbers, 01 natural sciences, Mathematics, Pell number, Linear form in logarithms
Abstract: In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2.
Published: 2019
Full Text: View/download PDF

18. An elementary property of correlations

Author: Giovanni Coppola, Università degli Studi di Salermo, and Università degli Studi di Salerno (UNISA)
Subjects: [ MATH ] Mathematics [math], Pure mathematics, Mathematics::General Mathematics, Mathematics::Number Theory, Mathematics::Classical Analysis and ODEs, Lambda, Convolution, Ramanujan's sum, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, 11N37, 11N05, 11P32, 11N37, FOS: Mathematics, Number Theory (math.NT), [MATH]Mathematics [math], 11P32, Mathematics, Conjecture, Von Mangoldt function, Mathematics - Number Theory, Mathematics::History and Overview, Elementary property, shift Ramanujan expansion, 2k-twin primes 2010 Mathematics Subject Classification 11N05, Shift factor, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], correlation, symbols
Abstract: We study the shift-Ramanujan expansion (see 1705.07193) of general $f,g$ satisfying Ramanujan Conjecture, in order to get formulae, for their shifted convolution sum, say $C_{f,g}(N,a)$, of length $N$ and shift $a$ (so, the Ramanujan expansion is with respect to a>0). We prove that, assuming Delange Hypothesis (DH) for the expansion, we get say Ramnujan exact explicit formula (R.e.e.f.). A noteworthy case, of course, is $f=g=\Lambda$, the von Mangoldt function, so $C_{\Lambda,\Lambda}(N,2k)$, for natural $k$, regards $2k-$twin primes; assuming $(DH)$ for them, we get (from corresponding R.e.e.f.) the proof, easily, of Hardy-Littlewood Conjecture for them., Comment: Assuming Delange Hypothesis(DH), we prove the "Ramanujan exact explicit formula" for $f,g$ correlation; for 2k-twin primes, assuming $(DH)$ we prove Hardy-Littlewood Conjecture
Published: 2019

19. On certain sums over ordinates of zeta-zeros II

Author: Kristian Seip, Eero Saksman, Aleksandar Ivić, Andriy Bondarenko, Norwegian University of Science and Technology [Trondheim] (NTNU), Norwegian University of Science and Technology (NTNU), and Serbian Academy of Sciences and Arts (SASA)
Subjects: [ MATH ] Mathematics [math], Mathematics - Number Theory, Laurent series, Analytic continuation, Second moment of area, Function (mathematics), second moment, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, Riemann hypothesis, symbols.namesake, analytic continuation, 11M06, Critical line, symbols, FOS: Mathematics, Number Theory (math.NT), Laurent expansion, [MATH]Mathematics [math], 2010 Mathematics Subject Classification 11M06 Keywords Riemann zeta-function, Mathematics
Abstract: Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001., Comment: 12 pages
Published: 2019

20. A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences

Author: Jean-Marc Deshouillers, Institut de Mathématiques de Bordeaux (IMB), 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: [ MATH ] Mathematics [math], Segal-Piatetski-Shapiro sequences, media_common.quotation_subject, 010102 general mathematics, Cube (algebra), 010103 numerical & computational mathematics, Infinity, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, 11N37, 11L07, discrepancy 2010 Mathematics Subject Classification 11B75, cube-free numbers, estimation of trigonometric sums, 0101 mathematics, [MATH]Mathematics [math], 11N56, 11L03, Mathematics, media_common
Abstract: Using a method due to G. J. Rieger, we show that for $1 < c < 2 $ one has, as $x$ tends to infinity $$\textrm{Card}{n \leq x : \lfloor{n^c}\rfloor} \ \textrm{ is cube-free} } = \frac{x}{\zeta(3)} + O (x^{ (c+1)/3} \log x)$$ , thus improving on a recent result by Zhang Min and Li Jinjiang.
Published: 2019

21. Explicit abc-conjecture and its applications

Author: Chi Chim Kwok, Saranya G. Nair, and Tarlok Nath Shorey
Subjects: Discrete mathematics, [ MATH ] Mathematics [math], Factorial, Conjecture, factorials, abc conjecture, State (functional analysis), Astrophysics::Cosmology and Extragalactic Astrophysics, explicit conjecture, 11B25, 11Z05, Statistics::Computation, Primes, 11D41, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Product (mathematics), Diophantine equations 2010 Mathematics Subject Classification 11A41, abc-conjecture, High Energy Physics::Experiment, Mathematics, 11N13
Abstract: We state well-known abc-conjecture of Masser-Oesterlé and its explicit version, popularly known as the explicit abc-conjecture, due to Baker. Laishram and Shorey derived from the explicit abc-conjecture that (1.1) implies that $c < N^{1.75}$. We give a survey on improvements of this result and its consequences. Finally we prove that $c < N^{1.7}$ and apply this estimate on an equation related to a conjecture of Hickerson that a factorial is not a product of factorials non-trivially.
Published: 2019
Full Text: View/download PDF

22. Fluctuation of the primitive of Hardy's function

Author: Matti Jutila
Subjects: [ MATH ] Mathematics [math], Mathematics::Functional Analysis, S function, Riemann's zeta-function, Mathematics::Classical Analysis and ODEs, Function (mathematics), Statistics::Computation, Gibbs phenomenon, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, symbols, the Gibbs phenomenon 2010 Mathematics Subject Classification 11M06, Hardy's function, Mathematics, Mathematical physics
Abstract: We consider oscillatory properties of the primitive of Hardy's function including a certain Gibbs phenomenon due to M. A. Korolev.
Published: 2019
Full Text: View/download PDF

23. The Barban-Vehov Theorem in Arithmetic Progressions

Author: V. Kumar Murty
Subjects: [ MATH ] Mathematics [math], Mean square, Discrete mathematics, Sequence, Sieve (category theory), Upper and lower bounds, Brun-Titchmarsh theorem, 11N69, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Secondary 11B25, Arithmetic progression, Asymptotic formula, arithmetic progressions 2010 Mathematics Subject Classification Primary 11N37, Selberg's sieve, 11N13, 11N35, Brun–Titchmarsh theorem, Mathematics
Abstract: A result of Barban-Vehov (and independently Motohashi) gives an estimate for the mean square of a sequence related to Selberg's sieve. This upper bound was refined to an asymptotic formula by S. Graham in 1978. In 1992, I made the observation that Graham's method can be used to obtain an asymptotic formula when the sum is restricted to an arithmetic progression. This formula immediately gives a version of the Brun-Titchmarsh theorem. I am taking the occasion of a volume in honour of my friend S. Srinivasan to revisit and publish this observation in the hope that it might still be of interest.
Published: 2019
Full Text: View/download PDF

24. On an identity of Ramanujan

Author: Yoichi Motohashi
Subjects: [ MATH ] Mathematics [math], Pure mathematics, Ramanujan's identity, Mathematics::General Mathematics, Mathematics::Number Theory, media_common.quotation_subject, Selberg's ${\Lambda}^2$-sieve, Contrast (music), Mathematical proof, Ramanujan's sum, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, Identity (philosophy), symbols, Euler's formula, Mobius inversion, Möbius inversion, Mathematics, media_common
Abstract: Proofs published so far in articles and books, of the Ramanujan identity presented in this note, which depend on Euler products, are essentially the same as Ramanujan's original proof. In contrast, the proof given here is short and independent of the use of Euler products.
Published: 2019
Full Text: View/download PDF

25. On twists of smooth plane curves

Author: Eslam Badr, Francesc Bars, Elisa Lorenzo García, Universitat Autònoma de Barcelona [Barcelona] ( UAB ), Universiteit Leiden [Leiden], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Universitat Autònoma de Barcelona (UAB), Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), MTM2013-40680-P, Ministerio de Educación, Cultura y Deporte, Universiteit Leiden, 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), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: Surface (mathematics), Pure mathematics, Plane curve, Non-singular plane curves, Closure (topology), automorphism groups, Field (mathematics), 0102 computer and information sciences, 01 natural sciences, Separable space, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, Diagonal matrix, FOS: Mathematics, Automorphism groups, Number Theory (math.NT), 0101 mathematics, Twist, Algebraic Geometry (math.AG), Mathematics, Algebra and Number Theory, Mathematics - Number Theory, 11G30, 11D41, 14H37, 14H50, 14H45, 12F12, Plane (geometry), Applied Mathematics, 010102 general mathematics, 16. Peace & justice, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], non-singular plane curves, Computational Mathematics, 010201 computation theory & mathematics, twist, 11G30, 11D41,14H37, 14H50, 14H45, 12F12, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: International audience; Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\overline{k}$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over $k$ for its twists. We characterize twists possessing such models and use such characterization to improve, for the particular case of smooth plane curves, the algorithm to compute twists of non-hyperelliptic curves wrote recently down by the third author. We also show an example of a twist not admitting such non-singular plane model. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to compute all the twists of smooth plane curves with cyclic automorphism group having a $k$-model whose automorphism group is generated by a diagonal matrix. Some examples are also provided.
Published: 2019
Full Text: View/download PDF

26. Factorization of bivariate sparse polynomials

Author: Martín Sombra, Francesco Amoroso, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU), PICS 6381 ``Diophantine geometry and computer algebra'. MINECO research project MTM2015-65361-P, Laboratoire de Mathématiques Nicolas Oresme ( LMNO ), Université de Caen Normandie ( UNICAEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), 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 ), ICREA \& Universitat de Barcelona, Institució Catalana de Recerca i Estudis Avançats [Barcelona] ( ICREA ), and Universitat de Barcelona ( UB ) -Fundació Catalana per a la Recerca i la Innovació (FCRI)-ICREA-Universitat de Barcelona ( UB ) -Fundació Catalana per a la Recerca i la Innovació (FCRI)-ICREA
Subjects: Rational number, Cossos algebraics, Algebra and Number Theory, Conjecture, Generalization, Mathematics::Number Theory, Univariate, Bivariate analysis, Mathematical proof, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), Polynomials, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Àlgebra commutativa, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Factorization, FOS: Mathematics, Primary 13P05, Secondary 12Y05, Polinomis, Computer Science::Symbolic Computation, Commutative algebra, Algebraic fields, Function field, Mathematics
Abstract: We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a variant of the toric Bertini's theorem due to Zannier and Fuchs, Mantova and Zannier., Comment: 16 pages
Published: 2019
Full Text: View/download PDF

27. Counting points on genus-3 hyperelliptic curves with explicit real multiplication

Author: Simon Abelard, Pierre-Jean Spaenlehauer, Pierrick Gaudry, Cryptology, arithmetic : algebraic methods for better algorithms (CARAMBA), Department of Algorithms, Computation, Image and Geometry (LORIA - ALGO), Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Department of Algorithms, Computation, Image and Geometry (LORIA - ALGO), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Cryptology, arithmetic : algebraic methods for better algorithms ( CARAMBA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Department of Algorithms, Computation, Image and Geometry ( LORIA - ALGO ), Laboratoire Lorrain de Recherche en Informatique et ses Applications ( LORIA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Laboratoire Lorrain de Recherche en Informatique et ses Applications ( LORIA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), and Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)
Subjects: FOS: Computer and information sciences, Computer Science - Symbolic Computation, [INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC], Endomorphism, Mathematics::Number Theory, 010103 numerical & computational mathematics, Symbolic Computation (cs.SC), 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, symbols.namesake, FOS: Mathematics, Number Theory (math.NT), Cubic field, 0101 mathematics, Algebraic Geometry (math.AG), Mathematics, Discrete mathematics, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Ring (mathematics), Mathematics - Number Theory, 010102 general mathematics, Order (ring theory), 16. Peace & justice, [ INFO.INFO-SC ] Computer Science [cs]/Symbolic Computation [cs.SC], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Riemann zeta function, [ INFO.INFO-CC ] Computer Science [cs]/Computational Complexity [cs.CC], Finite field, symbols, Multiplication, Hyperelliptic curve
Abstract: We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[\eta]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\widetilde O((\log q)^6)$ bit-operations, where the constant in the $\widetilde O()$ depends on the ring $\mathbb Z[\eta]$ and on the degrees of polynomials representing the endomorphism $\eta$. As a proof-of-concept, we compute the zeta function of a curve defined over a 64-bit prime field, with explicit real multiplication by $\mathbb Z[2\cos(2\pi/7)]$., Comment: Proceedings of the ANTS-XIII conference (Thirteenth Algorithmic Number Theory Symposium)
Published: 2018

28. Annihilation of tor\_p(G\_S^ab) for real abelian extensions

Author: Gras , Georges, Laboratoire de Mathématiques de Besançon, Chercheur indépendant, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], class field theory, annihilation of p-torsion modules, cyclotomic units, Mathematics - Number Theory, p-adic L-functions, 11R37, 11R29, 11R42, 11S15, Mathematics::Number Theory, Stickelberger's elements, FOS: Mathematics, Number Theory (math.NT), abelian p-ramification, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract: Preprint of a paper to appear in "Communications in Advanced Mathematical Sciences". Let K be a real abelian extension of Q. Let p be a prime number, S the set of p-places of K and G\_K,S the Galois group of the maximal S-ramified pro-p-extension of K (i.e., unramified outside p and infinity). We revisit the problem of annihilation of the p-torsion group T\_K:=tor\_Z\_p(G\_K,S^ab) initiated by us and Oriat then systematized in our paper on the construction of p-adic L-functions in which we obtained a canonical ideal annihilator of T\_K in full generality (1978--1981). Afterwards (1992--2014) some annihilators, using cyclotomic units, were proposed by Solomon, Belliard--Nguyen Quang Do, Nguyen Quang Do--Nicolas, All, Belliard--Martin.In this text, we improve our original papers and show that, in general, the Solomon elements are not optimal and/or partly degenerated. We obtain, whatever K and p, an universal non-degenerated annihilator in terms of p-adic logarithms of cyclotomic numbers related to L\_p-functions at s=1 of its primitive characters of K (Theorem 9.4). Some computations are given with PARI programs; the case p=2 is analyzed and illustrated in degrees 2, 3, 4 to test a conjecture., Analyse (by means of PARI programs) of the annihilations of "real classes" given by the Solomon elements and a new annihilator. Communications in Advanced Mathematical Sciences, Emrah Evren Kara, In press
Published: 2018
Full Text: View/download PDF

29. Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

Author: Pierrick Gaudry, Pierre-Jean Spaenlehauer, Simon Abelard, Cryptology, arithmetic : algebraic methods for better algorithms ( CARAMBA ), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Department of Algorithms, Computation, Image and Geometry ( LORIA - ALGO ), Laboratoire Lorrain de Recherche en Informatique et ses Applications ( LORIA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Laboratoire Lorrain de Recherche en Informatique et ses Applications ( LORIA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), Cryptology, arithmetic : algebraic methods for better algorithms (CARAMBA), Department of Algorithms, Computation, Image and Geometry (LORIA - ALGO), Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Department of Algorithms, Computation, Image and Geometry (LORIA - ALGO), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)
Subjects: Computer Science - Symbolic Computation, FOS: Computer and information sciences, Polynomial, 010103 numerical & computational mathematics, 0102 computer and information sciences, Symbolic Computation (cs.SC), 01 natural sciences, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, Genus (mathematics), FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Algebraic Geometry (math.AG), Local zeta-function, Mathematics, Discrete mathematics, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Spacetime, Mathematics - Number Theory, Applied Mathematics, Numerical analysis, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ INFO.INFO-SC ] Computer Science [cs]/Symbolic Computation [cs.SC], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Computational Mathematics, Computational Theory and Mathematics, 010201 computation theory & mathematics, Las Vegas algorithm, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Constant (mathematics), Hyperelliptic curve, Analysis
Abstract: We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems. Our main result improves on previously known complexity bounds by showing that there exists a constant $c>0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O((\log q)^{cg})$ as $q$ grows and the characteristic is large enough., To appear in Foundations of Computational Mathematics
Published: 2018
Full Text: View/download PDF

30. Chevalley–Weil theorem and subgroups of class groups

Author: Yuri Bilu, Jean Gillibert, 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), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-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 Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), 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 ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), and Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: Large class, Pure mathematics, Class (set theory), Mathematics - Number Theory, Group (mathematics), General Mathematics, 010102 general mathematics, Counting Number, Extension (predicate logic), Algebraic number field, 16. Peace & justice, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics::K-Theory and Homology, 0103 physical sciences, FOS: Mathematics, Cover (algebra), 010307 mathematical physics, Number Theory (math.NT), 0101 mathematics, Algebra over a field, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: We prove, under some mild hypothesis, that an \'etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an "absolute" version of the Chevalley-Weil theorem. Using this result, we are able to generalize the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group., Comment: Minor improvements following the referee's suggestions. To appear in Israel J. of Math
Published: 2018
Full Text: View/download PDF

31. Solving the Riemann Hypothesis with Green's Function and a Gelfand Triplet

Author: Moxley , Frederick and Louisiana State University [Baton Rouge] ( LSU )
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], [ PHYS ] Physics [physics], [ MATH.MATH-QA ] Mathematics [math]/Quantum Algebra [math.QA], [ PHYS.QPHY ] Physics [physics]/Quantum Physics [quant-ph], [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph]
Abstract: The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a Hermitian Hamiltonian that maps to the nontrivial zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results using Green's functions. A Gelfand triplet is then used to ensure that the eigenvalues are well defined. The holomorphicity of the resulting eigenvalues is demonstrated, and it is shown that that the expectation value of the Hamiltonian operator is also zero such that the nontrivial zeros of the Riemann zeta function are not observable. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function exists at σ = 1/2, and a general solution is obtained by performing an invariant similarity transformation.
Published: 2018

32. Proposition de démonstration de la conjecture de Goldbach

Author: Vella-Chemla, Denise, Chercheur indépendant, and Chemla, Denise
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract: This note is a proposition for a Goldbach's conjecture demonstration based on a counting of types of decompositions of even numbers using a four letters language., Cette note est une proposition de démonstration de la conjecture de Goldbach basée sur un comptage des types de décompositions de nombres pairs en sommes de deux impairs et qui utilise un langage à 4 lettres.
Published: 2018

33. A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE $q$-GAMMA FUNCTION

Author: Qi , Feng, Henan Polytechnic University, Tianjin Polytechnic University, Inner Mongolia University for Nationalities, University of Electronic Science and Technology of China ( UESTC ), and University of Electronic Science and Technology of China (UESTC)
Subjects: inequality, [ MATH.MATH-CA ] Mathematics [math]/Classical Analysis and ODEs [math.CA], logarithmically completely monotonic function, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], [ MATH.MATH-CO ] Mathematics [math]/Combinatorics [math.CO], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], MSC: Primary 44A10, Secondary 05A20, 26A48, 26D07, 33B15, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], $q$-gamma function, $q$-multinomial coefficient, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Statistics::Methodology, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], $q$-multivariate beta function, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract: In the paper, the author proves the logarithmically complete monotonicity of a function involving the q-gamma function and derives some inequalities for the $q$-multinomial coefficient and the q-multivariate beta function. These conclusions generalize corresponding ones for the gamma function, the multinomial coefficient, and the multivariate beta function respectively.
Published: 2018

34. A proposition for a Goldbach's conjecture demonstration

Author: Vella-Chemla , Denise and Chercheur indépendant
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]
Abstract: We try to demonstrate Goldbach's conjecture. Let us define 4 variables : X a (n) = #{p + q = n such that p and q odd, 3 p n/2, p and q prime} X b (n) = #{p + q = n such that p and q odd, 3 p n/2, p compound and q prime} X c (n) = #{p + q = n such that p and q odd, 3 p n/2, p prime and q compound} X d (n) = #{p + q = n such that p and q odd, 3 p n/2, p and q compound} In the following, let us note E(x) the integer part of x (i.e. ⌊x⌋). One will assume that X a (2n) = 0 while X a (n) is not zero and try to end up at a contradiction.; This note is a proposition for a Goldbach's conjecture demonstration based on counting decompositions of even numbers using a four letters language.
Published: 2018

35. On the Prym variety of genus 3 covers of genus 1 curves

Author: Christophe Ritzenthaler, Matthieu Romagny, Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 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 Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), and Guillemer, Marie-Annick
Subjects: Pure mathematics, Field (mathematics), 010103 numerical & computational mathematics, Prym variety, 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, 14Q05, Genus (mathematics), FOS: Mathematics, Number Theory (math.NT), plane quartics, 0101 mathematics, Algebraic Geometry (math.AG), ComputingMilieux_MISCELLANEOUS, Mathematics, singular covers, Algebra and Number Theory, Mathematics - Number Theory, 14Q05, 14H40, 11G05, 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Genus 3 curves, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Product (mathematics), Cover (algebra), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Geometry and Topology, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract: Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin., Comment: Published version
Published: 2018
Full Text: View/download PDF

36. Groestl Distinguishing Attack: A New Rebound Attack of an AES-like Permutation

Author: Cauchois, Victor, Gomez, Clément, Lercier, Reynald, DGA Maîtrise de l'information (DGA.MI), Direction générale de l'Armement (DGA), 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), DGA Maîtrise de l'information ( DGA.MI ), Délégation Générale de l'Armement, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Lercier, Reynald
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], [INFO.INFO-CR]Computer Science [cs]/Cryptography and Security [cs.CR], [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], [ INFO.INFO-CR ] Computer Science [cs]/Cryptography and Security [cs.CR], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [INFO.INFO-CR] Computer Science [cs]/Cryptography and Security [cs.CR], [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: International audience
Published: 2018

37. Reduction type of non-hyperelliptic genus 3 curves

Author: Lercier , Reynald, García , Elisa Lorenzo, Ritzenthaler , Christophe, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Marseille ( I2M ), and Aix Marseille Université ( AMU ) -Ecole Centrale de Marseille ( ECM ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: smooth plane quartic, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics::Algebraic Geometry, 11G20, 14Q05, 14D10, 14D20, 14H25, hyperelliptic, reduction, valuation, invarian ts, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: 26 pages; Let $C/K: F=0$ be a smooth plane quartic over a complete discrete valuation field $K$ whose residue field is of characteristic different from $2,3, 5$ and $7$. We give various characterizations of the reduction (i.e. non-perelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of $C$: in terms of the existence of a particular plane quartic model; in terms of the valuations of the theta constants of $C$; in terms of the valuations of the Dixmier-Ohno invariants of $C$. The last one leads to an easy computable criterion.
Published: 2018

38. Affine smooth group schemes over the dual numbers

Author: ROMAGNY , Matthieu, Tossici , Dajano, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bordeaux ( IMB ), 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: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT]
Abstract: We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 → Lie(G, I) → E → G → 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k ⊕ I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $O_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.
Published: 2018

39. ZpL: a p-adic precision package

Author: Xavier Caruso, David Roe, Tristan Vaccon, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Massachusetts Institute of Technology ( MIT ), Mathématiques & Sécurité de l'information ( XLIM-MATHIS ), XLIM ( XLIM ), Université de Limoges ( UNILIM ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Limoges ( UNILIM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Massachusetts Institute of Technology (MIT), Mathématiques & Sécurité de l'information (XLIM-MATHIS), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), 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, and 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)
Subjects: FOS: Computer and information sciences, Computer Science - Symbolic Computation, [ INFO.INFO-NA ] Computer Science [cs]/Numerical Analysis [cs.NA], Automatic differentiation, Differential equation, p-adic precision, 010103 numerical & computational mathematics, Symbolic Computation (cs.SC), Tracking (particle physics), 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Automatic Differentiation, Mathematical software, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Commutative algebra, Ultrametric space, Mathematics, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Mathematics - Number Theory, Computer Science - Numerical Analysis, Numerical Analysis (math.NA), [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ INFO.INFO-SC ] Computer Science [cs]/Symbolic Computation [cs.SC], Large sample, 010101 applied mathematics, Algebra, Linear algebra, Algorithms
Abstract: International audience; We present a new package ZpL for the mathematical software system SageMath. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in previous papers by the same authors. The underlying algorithms are mostly based on automatic differentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the benefits of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, commutative algebra and differential equations.
Published: 2018
Full Text: View/download PDF

40. Odd zeta motive and linear forms in odd zeta values

Author: Clément Dupont, Max-Planck-Institut für Mathematik ( MPI ), Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Max-Planck-Institut für Mathematik (MPI), Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Algebra and Number Theory, Mathematics - Number Theory, Series (mathematics), Mathematics::Number Theory, 010102 general mathematics, Structure (category theory), Parity (physics), 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Interpretation (model theory), [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, Simple (abstract algebra), 0103 physical sciences, 010307 mathematical physics, Integral formula, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $\zeta(n)$. They naturally include the Beukers-Rhin-Viola integrals for $\zeta(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series., Comment: With a joint appendix with Don Zagier. Main results unchanged; minor changes in the presentation; corrected the proof of Prop. 3.12; now matches the published version; 31 pages
Published: 2018
Full Text: View/download PDF

41. Codes from unit groups of division algebras over number fields

Author: Christian Maire, Aurel Page, Laboratoire de Mathématiques de Besançon ( LMB ), Centre National de la Recherche Scientifique ( CNRS ) -Université de Franche-Comté ( UFC ), 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 ), Region Bourgogne Franche-Comté. EPSRC financial support via the EPSRC Programme Grant EP/K034383/1 LMF: L-functions and modular forms, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), 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), ANR-17-CE40-0012,FLAIR,Familles de fonctions L: analyse, interactions, résultats effectifs(2017), ANR-17-EURE-0002,EIPHI,Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité(2017), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-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)-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: FOS: Computer and information sciences, General Mathematics, Computer Science - Information Theory, Algebraic geometry, [ MATH.MATH-IT ] Mathematics [math]/Information Theory [math.IT], 01 natural sciences, Division Algebras over Number Fields, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], 0103 physical sciences, FOS: Mathematics, 2000 Mathematics Subject Classification - 11T71, 94B40, 11R52, 94B75, sum-rank metric, Number Theory (math.NT), 0101 mathematics, Quaternion, Mathematics, Number Field Codes, Discrete mathematics, Degree (graph theory), Mathematics - Number Theory, Information Theory (cs.IT), 010102 general mathematics, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], Algebraic number field, Division (mathematics), Noncommutative geometry, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Metric (mathematics), Asymptotically Good Codes, 010307 mathematical physics, Unit (ring theory)
Abstract: International audience; Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.
Published: 2018
Full Text: View/download PDF

42. LA PREUVE DE LA CONJECTURE ABC

Author: Sghiar, Mohamed, Chercheur indépendant, and Sghiar, Mohamed
Subjects: la théorie des nombres, le théorème de Faltings, [ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM], la conjecture de Pillai, conjecture de Mordell, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], [MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA], les premiers de Wieferich, le théorème de Fermat-Wiles, le théorème de Baker, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], la conjecture d'Erdős-Woods, le théorème de Roth, équation de Brocard, la conjecture abc, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], le théorème de Bombieri-Vinogradov
Abstract: The purpose of this article is to demonstrate the abc conjecture and to deduce among other things a simple proof of the Fermat-Wiles theorem and a proof of the Pillai conjecture., Le but de cet article est de démontrer la conjecture abc et d'en déduire entre autres une preuve simple du théorème de Fermat-Wiles et une preuve de la conjecture de Pillai.
Published: 2018

43. Polynômes de Ore en une variable

Author: Caruso, Xavier, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -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), 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), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 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 Caruso, Xavier
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-RA ] Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract: Les polynômes de Ore sont une variante non commutative des polynômes classiques qui interviennent en algèbre semi-linéaire et dans l'´ etude deséquationsdeséquations différentielles linéaires de la mêmemanì ere que les polynômes usuels interviennent en algèbre linéaire (polynômes d'endomorphisme, polynômes caractéristiques, etc.) En particulier, la factorisation des po-lynômes de Ore est liée de très prèsprès`prèsà la réduction des endomorphismes semi-linéaires etàet`età celle deséquationsdeséquations différentielles linéaires. Le but de ce cours est définir les polynômes de Ore, d'´ etablir leurs principales propriétés arithmétiques et d'´ etudier leur factorisation. Nous mettons en place tout un arnesal théorique, centrée sur la notion d'algèbre d'Azumaya, qui permet, dans certains cas, d'obtenir des théorèmes de structure donnant des renseignements très précis sur la forme des factorisations des polynômes de Ore. Le cours est illustré de nombreux exemples.
Published: 2018

44. Généralisation d'un théorème de Greenberg

Author: Jaulent , Jean-François, 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 ), Institut de Mathématiques de Bordeaux (IMB), 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: Greenberg's theorem, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Iwasawa Theory, 11R23, Cyclotomic conjecture, Gross-Kuz'min conjecture, Leopoldt's conjecture, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Kuz'min's isomorphism
Abstract: International audience; We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic Z ℓ-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the classical conjectures of Leopoldt and of Gross-Kuz'min. We so extend a result of semi-simplicity of Greenberg and, by the way, an isomorphism of Kuz’min.; Nous formulons une conjecture générale sur le polynôme caractéristique des modules d'Iwasawa S-décomposés T-ramifiés au-dessus de la Z ℓ-extension cyclotomique d'un corps de nombres. Nous montrons qu'elle est en fait équivalente à la conjonction des conjectures de Leopoldt et de Gross-Kuz'min. Nous étendons ainsi un théorème de semi-simplicité de Greenberg et, au passage, un isomorphisme de Kuz’min.
Published: 2018

45. Nori motives of curves with modulus and Laumon 1-motives

Author: Florian Ivorra, Takao Yamazaki, 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), Institute of Mathematics, Tohoku University [Sendai], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -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: Pure mathematics, General Mathematics, Modulus, Disjoint sets, 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Mathematics::Category Theory, 0103 physical sciences, [ MATH.MATH-KT ] Mathematics [math]/K-Theory and Homology [math.KT], FOS: Mathematics, curves with modulus, Number Theory (math.NT), 19E15, 16G20, 14F42, 0101 mathematics, Algebraic Geometry (math.AG), Mathematics, 14F42, Mathematics - Number Theory, 010102 general mathematics, Quiver, Representation (systemics), K-Theory and Homology (math.KT), Algebraic number field, 16. Peace & justice, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Quiver representation, Motives, Mathematics - K-Theory and Homology, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], 010307 mathematical physics, Abelian category, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: Let $k$ be a number field. We describe the category of Laumon 1-isomotives over $k$ as the universal category in the sense of Nori associated with a quiver representation built out of smooth proper $k$-curves with two disjoint effective divisors and a notion of $H^1_\dR$ for such "curves with modulus". This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives., Comment: 30 pages. The main result is now valid over any number field
Published: 2018
Full Text: View/download PDF

46. Un calcul d'anneaux de déformations potentiellement Barsotti--Tate

Author: Agnès David, Xavier Caruso, Ariane Mézard, Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université du Luxembourg (Uni.lu), GLIMM-3988642, Fonds National de la Recherche Luxembourg, 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), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université du Luxembourg ( Uni.lu ), IMJ - Théorie des nombres, Institut de Mathématiques de Jussieu ( IMJ ), and Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: Pure mathematics, variétés de Kisin, General Mathematics, Mathematics::Number Theory, Structure (category theory), Type (model theory), 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Corollary, espaces de déformations, 0103 physical sciences, 0101 mathematics, 11F80, 11S37, Mathematics, Conjecture, Mathematics - Number Theory, Degree (graph theory), Applied Mathematics, 010102 general mathematics, Absolute Galois group, Extension (predicate logic), 16. Peace & justice, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Algebra, 010307 mathematical physics, Variety (universal algebra), conjecture de Breuil-Mézard, représentations galoisiennes p-adiques
Abstract: Let F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in the particular case where F has degree 2 over Q_p and determine this way almost all these deformations rings. In this particular case, we observe a close relationship between the structure of these deformations rings and the geometry of the associated Kisin variety. As a corollary and still assuming that F has degree 2 over Q_p, we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or 1., Comment: 57 pages, in French
Published: 2018
Full Text: View/download PDF

47. Sato–Tate distributions of twists of the Fermat and the Klein quartics

Author: Francesc Fité, Andrew V. Sutherland, Elisa Lorenzo García, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics [MIT], Massachusetts Institute of Technology ( MIT ), Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Massachusetts Institute of Technology (MIT), DMS-1522526, National Science Foundation, 550033, Simons Foundation, 682152, H2020 European Research Council, CRC 701, German Research Council, MDM-2014-0445, Excellence Program Maria de Maeztu, 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), and Sutherland II, Andrew Victor
Subjects: Pure mathematics, Asymptotic distribution, 010103 numerical & computational mathematics, 01 natural sciences, Theoretical Computer Science, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, Mathematics (miscellaneous), Mathematics::Algebraic Geometry, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Abelian group, Mathematics, Fermat's Last Theorem, Mathematics - Number Theory, Applied Mathematics, 010102 general mathematics, Cube (algebra), Limiting, Algebraic number field, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Computational Mathematics, Elliptic curve, 14G10 (primary) 14H45, 14K15, 11G10, 11G40 (secondary), Euler's formula, symbols
Abstract: We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is [bar over Q]-isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato–Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients., National Science Foundation (U.S.) (Grant DMS-1115455), National Science Foundation (U.S.) (Grant DMS-1522526)
Published: 2018
Full Text: View/download PDF

48. The Picard group of Brauer-Severi varieties

Author: Francesc Bars, Eslam Badr, Elisa Lorenzo García, Universitat Autònoma de Barcelona (UAB), Institut de Recherche Mathématique de Rennes (IRMAR), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Universitat Autònoma de Barcelona [Barcelona] ( UAB ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 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, and 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)
Subjects: Pure mathematics, Automorphisms, theory of twists, automorphisms, Plane curve, Picard groups, picard groups, General Mathematics, Picard group, 010103 numerical & computational mathematics, Base field, 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics::Algebraic Geometry, Mathematics::K-Theory and Homology, QA1-939, FOS: Mathematics, cyclic algebras, Brauer-Severi varieties, Number Theory (math.NT), 0101 mathematics, Mathematics::Representation Theory, Mathematics, 11d41, 14j50, Mathematics - Number Theory, brauer-severi varieties, 010102 general mathematics, Mathematics::Rings and Algebras, Theory of Twists, Automorphism, 14j70, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 11r34, Cyclic Algebras, 11g35, 14j26
Abstract: In this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular,we provide such generators for all Brauer-Severi surfaces. To produce these generators we use the theory of twists of smooth plane curves.
Published: 2018
Full Text: View/download PDF

49. La preuve de la conjecture de Birch et Swinnerton-Dyer

Author: Sghiar , Mohamed, Chercheur indépendant, and Sghiar, Mohamed
Subjects: L-function, [ MATH.MATH-GR ] Mathematics [math]/Group Theory [math.GR], [ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM], Birch, Swinnerton-Dyer, [ INFO.INFO-CR ] Computer Science [cs]/Cryptography and Security [cs.CR], [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], [ MATH.MATH-RA ] Mathematics [math]/Rings and Algebras [math.RA], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], courbe elliptique, [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], zêta de Riemann, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], [INFO.INFO-CR] Computer Science [cs]/Cryptography and Security [cs.CR]
Abstract: The purpose of this article is to demonstrate the Birch and Swinnerton-Dyer Conjecture: If L is the function associated to an elliptic curve, then the order of the zero of L at s = 1 is exactly the order of the curve. In particular, the curve admits an infinity of rational points if and only if L(1)=0, Le but de cet article est de démontrer la Conjecture de Birch et Swinnerton-Dyer : Si L est la fonction associée à une courbe elliptique, alors l'ordre d'annulation de la fonction L en s=1 est exactement l'ordre de la courbe. En particulier, la courbe admet une infinité de points rationnels si et seulement si L(1)=0
Published: 2018

50. ON ARTIN'S CONJECTURE: LINEAR SLICES OF DIAGONAL HYPERSURFACES

Author: Brüdern, Jörg, Robert, Olivier, Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Mathematisches Institut Georg-August-Universität Göttingen (MATHEMATISCHES INSTITUT GOETTINGEN), Mathematisches Institut-Georg-August-University [Göttingen]-Göttingen, Combinatoire, théorie des nombres (CTN), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Physics::Popular Physics, Mathematics::Group Theory, Mathematics - Number Theory, Mathematics::History and Overview, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Computer Science::Computers and Society, Computer Science::Databases
Abstract: Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane., Comment: Transactions of the American Mathematical Society, American Mathematical Society, In press
Published: 2018

Catalog

Books, media, physical & digital resources

See catalog results

Searchworks

Select search scope, currently: Articles Catalog books, media & more in Jio Institute collections Articles journal articles & other e-resources

Search

Search Constraints

Refine your results

Search Limiters

Topic

Publication Year Range

Language

Journal

Database

Publisher

201 results on '"[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]"'

Search Results

Catalog

Select search scope, currently: Articles

Catalog

books, media & more in Jio Institute collections

Articles

journal articles & other e-resources