47 research outputs found
Linear independence of powers of singular moduli of degree 3
We show that two distinct singular moduli , such that for
some positive integers the numbers and are
linearly dependent over generate the same number field of degree
at most . This completes a result of Riffaut, who proved the above theorem
except for two explicit pair of exceptions consisting of numbers of degree .
The purpose of this article is to treat these two remaining cases
Entre les lignes
This article is based on an investigation of regular swimmers in a Parisian pool. Firstly, it describes the method of the survey in a field of work where the researcher is also an actor. It shows how long-term and deep-rooted participation, as well as the daily sharing of the swimming activity, are two main assets of this « field work at home ». Secondly, this article focuses on the connection between swimmers and water as an object. Swimmers sought out water that was associated with a strong supportive imagery and with feelings of well-being, portage and inclusion. Finally, this article explores what motivates swimmers to practice their activity, which is often focused on a need to compensate for a physical problem or an existential issue.
Cet article est basĂ© sur une enquĂȘte menĂ©e auprĂšs de nageurs rĂ©guliers dâune piscine parisienne. Il dĂ©crit, dans un premier temps, les modalitĂ©s dâinscription du chercheur sur un terrain dans lequel il est lui-mĂȘme acteur. Il montre comment lâancrage dans la durĂ©e et le partage quotidien de lâactivitĂ© Ă©tudiĂ©e constituent deux atouts privilĂ©giĂ©s de ce « terrain chez soi ». Il sâintĂ©resse, dans un second temps, au rapport des nageurs Ă lâeau en tant que matiĂšre. Une eau associĂ©e Ă un imaginaire hautement porteur de sens et Ă des sensations de bien-ĂȘtre, de portage ou dâenveloppement, recherchĂ©s des nageurs. Il explore enfin les motivations des nageurs Ă pratiquer leur activitĂ©. Lesquelles sont souvent centrĂ©es autour dâun besoin de compenser un problĂšme physique ou une difficultĂ© existentielle.
 
Helping yourself by helping others : Volunteering : A space of self-construction and personal fulfillment
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Effective computation of special points
Ă partir du thĂ©orĂšme dâAndrĂ© en 1998, qui est la premiĂšre contribution non triviale Ă la conjecture de AndrĂ©-Oort sur les sous-variĂ©tĂ©s spĂ©ciales des variĂ©tĂ©s de Shimura, la principale problĂ©matique de cette thĂšse est dâĂ©tudier les propriĂ©tĂ©s diophantiennes des modules singuliers, en caractĂ©risant les points de multiplication complexe (x; y) satisfaisant un type dâĂ©quation donnĂ© de la forme F(x; y) = 0, pour un polynĂŽme irrĂ©ductible F(X; Y ) Ă coefficients complexes. Plus spĂ©cifiquement, nous traitons deux Ă©quations impliquant des puissances de modules singuliers. Dâune part, nous montrons que deux modules singuliers x; y tels que les nombres 1, xm et yn soient linĂ©airement dĂ©pendants sur Q, pour des entiers strictement positifs m; n, doivent ĂȘtre de degrĂ© au plus 2, ce qui gĂ©nĂ©ralise un rĂ©sultat dâAllombert, Bilu et Pizarro-Madariaga, qui ont Ă©tudiĂ© les points de multiplication complexe appartenant aux droites de C2 dĂ©finies sur Q. Dâautre part, nous montrons que, sauf cas âĂ©videntsâ, le produit de nâimporte quelles puissances entiĂšres de deux modules singuliers ne peut ĂȘtre un nombre rationnel non nul, ce qui gĂ©nĂ©ralise un rĂ©sultat de Bilu, Luca et Pizarro- Madariaga, qui ont ont Ă©tudiĂ© les points de multiplication complexe appartenant aux hyperboles xy = A, oĂč A 2 Qx. Les mĂ©thodes que nous dĂ©veloppons reposent en grande partie sur les propriĂ©tĂ©s des corps de classes engendrĂ©s par les modules singuliers, les estimations de la fonction j-invariant et les estimations des formes linĂ©aires logarithmiques. Nous dĂ©terminons Ă©galement les corps engendrĂ©s par les sommes et les produits de deux modules singuliers x et y : nous montrons que le corps Q(x; y) est engendrĂ© par la somme x + y, Ă moins que x et y soient conjuguĂ©s sur Q, auquel cas x + y engendre un sous-corps de degrĂ© au plus 2 ; le mĂȘme rĂ©sultat demeure pour le produit xy. Nos preuves sont assistĂ©es par le logiciel PARI/GP, que nous utilisons pour procĂ©der Ă des vĂ©rifications dans des cas particuliers explicites.Starting for AndrĂ©âs Theorem in 1998, which is the first non-trivial contribution to the celebrated AndrĂ©-Oort conjecture on the special subvarieties of Shimura varieties, the main purpose of this thesis is to study Diophantine properties of singular moduli, by characterizing CM-points (x; y) satisfying a given type of equation of the form F(x; y) = 0, for an irreducible polynomial F(X; Y ) with complex coefficients. More specifically, we treat two different equations involving powers of singular moduli. On the one hand, we show that two distinct singular moduli x; y such that the numbers 1, xm and yn are linearly dependent over Q, for some positive integers m; n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C2 defined over Q. On the other hand, we show that, with âobviousâ exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to hyperbolas xy = A, where A 2 Qx. The methods we develop lie mainly on the properties of ring class fields generated by singular moduli, on estimations of the j-function and on estimations of linear forms in logarithms. We also determine fields generated by sums and products of two singular moduli x and y : we show that the field Q(x; y) is generated by the sum x + y, unless x and y are conjugate over Q, in which case x + y generate a subfield of degree at most 2 ; the same holds for the product xy. Our proofs are assisted by the PARI/GP package, which we use to proceed to verifications in particular explicit cases
Fields generated by sums and products of singular moduli
We show that the field , generated by two singular
moduli~ and~, is generated by their sum , unless~ and~ are
conjugate over~, in which case generates a subfield of
degree at most~. We obtain a similar result for the product of two singular
moduli.Comment: Acknowledgment
Calcul effectif de points spéciaux
Starting for AndrĂ©âs Theorem in 1998, which is the first non-trivial contribution to the celebrated AndrĂ©-Oort conjecture on the special subvarieties of Shimura varieties, the main purpose of this thesis is to study Diophantine properties of singular moduli, by characterizing CM-points (x; y) satisfying a given type of equation of the form F(x; y) = 0, for an irreducible polynomial F(X; Y ) with complex coefficients. More specifically, we treat two different equations involving powers of singular moduli. On the one hand, we show that two distinct singular moduli x; y such that the numbers 1, xm and yn are linearly dependent over Q, for some positive integers m; n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C2 defined over Q. On the other hand, we show that, with âobviousâ exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to hyperbolas xy = A, where A 2 Qx. The methods we develop lie mainly on the properties of ring class fields generated by singular moduli, on estimations of the j-function and on estimations of linear forms in logarithms. We also determine fields generated by sums and products of two singular moduli x and y : we show that the field Q(x; y) is generated by the sum x + y, unless x and y are conjugate over Q, in which case x + y generate a subfield of degree at most 2 ; the same holds for the product xy. Our proofs are assisted by the PARI/GP package, which we use to proceed to verifications in particular explicit cases.Ă partir du thĂ©orĂšme dâAndrĂ© en 1998, qui est la premiĂšre contribution non triviale Ă la conjecture de AndrĂ©-Oort sur les sous-variĂ©tĂ©s spĂ©ciales des variĂ©tĂ©s de Shimura, la principale problĂ©matique de cette thĂšse est dâĂ©tudier les propriĂ©tĂ©s diophantiennes des modules singuliers, en caractĂ©risant les points de multiplication complexe (x; y) satisfaisant un type dâĂ©quation donnĂ© de la forme F(x; y) = 0, pour un polynĂŽme irrĂ©ductible F(X; Y ) Ă coefficients complexes. Plus spĂ©cifiquement, nous traitons deux Ă©quations impliquant des puissances de modules singuliers. Dâune part, nous montrons que deux modules singuliers x; y tels que les nombres 1, xm et yn soient linĂ©airement dĂ©pendants sur Q, pour des entiers strictement positifs m; n, doivent ĂȘtre de degrĂ© au plus 2, ce qui gĂ©nĂ©ralise un rĂ©sultat dâAllombert, Bilu et Pizarro-Madariaga, qui ont Ă©tudiĂ© les points de multiplication complexe appartenant aux droites de C2 dĂ©finies sur Q. Dâautre part, nous montrons que, sauf cas âĂ©videntsâ, le produit de nâimporte quelles puissances entiĂšres de deux modules singuliers ne peut ĂȘtre un nombre rationnel non nul, ce qui gĂ©nĂ©ralise un rĂ©sultat de Bilu, Luca et Pizarro- Madariaga, qui ont ont Ă©tudiĂ© les points de multiplication complexe appartenant aux hyperboles xy = A, oĂč A 2 Qx. Les mĂ©thodes que nous dĂ©veloppons reposent en grande partie sur les propriĂ©tĂ©s des corps de classes engendrĂ©s par les modules singuliers, les estimations de la fonction j-invariant et les estimations des formes linĂ©aires logarithmiques. Nous dĂ©terminons Ă©galement les corps engendrĂ©s par les sommes et les produits de deux modules singuliers x et y : nous montrons que le corps Q(x; y) est engendrĂ© par la somme x + y, Ă moins que x et y soient conjuguĂ©s sur Q, auquel cas x + y engendre un sous-corps de degrĂ© au plus 2 ; le mĂȘme rĂ©sultat demeure pour le produit xy. Nos preuves sont assistĂ©es par le logiciel PARI/GP, que nous utilisons pour procĂ©der Ă des vĂ©rifications dans des cas particuliers explicites
Equations with powers of singular moduli
We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(Ï), j(Ï âČ) such that the numbers 1, j(Ï) m and j(Ï âČ) n are linearly dependent over Q for some positive integers m, n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C 2 defined over Q. On the other hand, we show that, with " obvious " exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to an hyperbola xy = A, where A â Q
Louis Napoléon Président de la République française : [estampe]
RĂ©fĂ©rence bibliographique : De Vinck, 15468Appartient Ă lâensemble documentaire : Est19Vinc
Effective computation of special points
Ă partir du thĂ©orĂšme dâAndrĂ© en 1998, qui est la premiĂšre contribution non triviale Ă la conjecture de AndrĂ©-Oort sur les sous-variĂ©tĂ©s spĂ©ciales des variĂ©tĂ©s de Shimura, la principale problĂ©matique de cette thĂšse est dâĂ©tudier les propriĂ©tĂ©s diophantiennes des modules singuliers, en caractĂ©risant les points de multiplication complexe (x; y) satisfaisant un type dâĂ©quation donnĂ© de la forme F(x; y) = 0, pour un polynĂŽme irrĂ©ductible F(X; Y ) Ă coefficients complexes. Plus spĂ©cifiquement, nous traitons deux Ă©quations impliquant des puissances de modules singuliers. Dâune part, nous montrons que deux modules singuliers x; y tels que les nombres 1, xm et yn soient linĂ©airement dĂ©pendants sur Q, pour des entiers strictement positifs m; n, doivent ĂȘtre de degrĂ© au plus 2, ce qui gĂ©nĂ©ralise un rĂ©sultat dâAllombert, Bilu et Pizarro-Madariaga, qui ont Ă©tudiĂ© les points de multiplication complexe appartenant aux droites de C2 dĂ©finies sur Q. Dâautre part, nous montrons que, sauf cas âĂ©videntsâ, le produit de nâimporte quelles puissances entiĂšres de deux modules singuliers ne peut ĂȘtre un nombre rationnel non nul, ce qui gĂ©nĂ©ralise un rĂ©sultat de Bilu, Luca et Pizarro- Madariaga, qui ont ont Ă©tudiĂ© les points de multiplication complexe appartenant aux hyperboles xy = A, oĂč A 2 Qx. Les mĂ©thodes que nous dĂ©veloppons reposent en grande partie sur les propriĂ©tĂ©s des corps de classes engendrĂ©s par les modules singuliers, les estimations de la fonction j-invariant et les estimations des formes linĂ©aires logarithmiques. Nous dĂ©terminons Ă©galement les corps engendrĂ©s par les sommes et les produits de deux modules singuliers x et y : nous montrons que le corps Q(x; y) est engendrĂ© par la somme x + y, Ă moins que x et y soient conjuguĂ©s sur Q, auquel cas x + y engendre un sous-corps de degrĂ© au plus 2 ; le mĂȘme rĂ©sultat demeure pour le produit xy. Nos preuves sont assistĂ©es par le logiciel PARI/GP, que nous utilisons pour procĂ©der Ă des vĂ©rifications dans des cas particuliers explicites.Starting for AndrĂ©âs Theorem in 1998, which is the first non-trivial contribution to the celebrated AndrĂ©-Oort conjecture on the special subvarieties of Shimura varieties, the main purpose of this thesis is to study Diophantine properties of singular moduli, by characterizing CM-points (x; y) satisfying a given type of equation of the form F(x; y) = 0, for an irreducible polynomial F(X; Y ) with complex coefficients. More specifically, we treat two different equations involving powers of singular moduli. On the one hand, we show that two distinct singular moduli x; y such that the numbers 1, xm and yn are linearly dependent over Q, for some positive integers m; n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C2 defined over Q. On the other hand, we show that, with âobviousâ exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to hyperbolas xy = A, where A 2 Qx. The methods we develop lie mainly on the properties of ring class fields generated by singular moduli, on estimations of the j-function and on estimations of linear forms in logarithms. We also determine fields generated by sums and products of two singular moduli x and y : we show that the field Q(x; y) is generated by the sum x + y, unless x and y are conjugate over Q, in which case x + y generate a subfield of degree at most 2 ; the same holds for the product xy. Our proofs are assisted by the PARI/GP package, which we use to proceed to verifications in particular explicit cases