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X-ORIGINAL-URL:https://www.math.ens.psl.eu
X-WR-CALDESC:évènements pour Département de mathématiques et applications
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TZID:Europe/Paris
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DTSTART:20160327T010000
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DTSTART:20161030T010000
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DTSTART;TZID=Europe/Paris:20160520T140000
DTEND;TZID=Europe/Paris:20160520T150000
DTSTAMP:20260415T080334
CREATED:20160520T120000Z
LAST-MODIFIED:20211104T100826Z
UID:8283-1463752800-1463756400@www.math.ens.psl.eu
SUMMARY:Irreducibility of Polynomials over Number Fields is Diophantine
DESCRIPTION:We show that irreducibility of a polynomial in any number of variables over a number field is a diophantine condition\, i.e. captured by an existential formula. This generalises a previous result by Colliot-Thélène and Van Geel that the set of non-nth-powers is diophantine for any n. Our method is heavily based on the Brauer group\, originating from Poonen’s use of quaternion algebras as a technical tool for first-order definitions in number fields.
URL:https://www.math.ens.psl.eu/evenement/irreducibility-of-polynomials-over-number-fields-is-diophantine/
LOCATION:ENS Salle W
CATEGORIES:Variétés rationnelles
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BEGIN:VEVENT
DTSTART;TZID=Europe/Paris:20160520T151500
DTEND;TZID=Europe/Paris:20160520T161500
DTSTAMP:20260415T080334
CREATED:20160520T131500Z
LAST-MODIFIED:20211104T100826Z
UID:8284-1463757300-1463760900@www.math.ens.psl.eu
SUMMARY:The Lang-Vojta conjecture and smooth hypersurfaces over number fields.
DESCRIPTION:Siegel proved the finiteness of the set of solutions to the unit equation in a number ring\, i.e.\, for a number field K with ring of integers O\, the equation x+y=1 has only finitely many solutions in O*. That is\, reformulated in more algebro-geometric terms\, the hyperbolic curve P^1-{0\,1\,infinite} has only finitely many ‘integral points’. In 1983\, Faltings proved the Mordell conjecture generalizing Siegel’s theorem: a hyperbolic complex algebraic curve has only finitely many integral points. Inspired by Faltings’s and Siegel’s finiteness results\, Lang and Vojta formulated a general finiteness conjecture for ‘integral points’ on complex algebraic varieties: a hyperbolic complex algebraic variety has only finitely many ‘integral points’. In this talk we will start by explaining the Lang-Vojta conjecture and then proceed to prove some of its consequences for the arithmetic of homogeneous polynomials over number fields. This is joint work with Daniel Loughran.
URL:https://www.math.ens.psl.eu/evenement/the-lang-vojta-conjecture-and-smooth-hypersurfaces-over-number-fields/
LOCATION:ENS Salle W
CATEGORIES:Variétés rationnelles
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Europe/Paris:20160520T164500
DTEND;TZID=Europe/Paris:20160520T174500
DTSTAMP:20260415T080334
CREATED:20160520T144500Z
LAST-MODIFIED:20211104T100826Z
UID:8285-1463762700-1463766300@www.math.ens.psl.eu
SUMMARY:La composition de Gauss pour les points entiers primitifs de sphères\, en suivant\, partiellement\, Gunawan.
DESCRIPTION:Gauss a donné des formules pour le nombre de points entiers primitifs de la 2-sphère de rayon au carré égal à n. Ces formules sont en termes de nombres de classes d’anneaux quadratiques de discriminant étroitement liés à n. Cela mène à la question de savoir si ceci peut être expliqué par une action libre et transitive du groupe de Picard de cet anneau sur l’ensemble des tels points entiers primitifs à symétries globales SO_3(Z) près. Ceci est en effet le cas\, et cette action peut être explicitée. L’outil utilisé est la théorie des schémas en groupes sur Z\, ce qui est plus direct que la cohomologie galoisienne et les adèles\, et remarquablement élémentaire. En fait\, Gross et Bhargava demandent une telle approche dans leur article ‘Arithmetic invariant theory’.
URL:https://www.math.ens.psl.eu/evenement/la-composition-de-gauss-pour-les-points-entiers-primitifs-de-spheres-en-suivant-partiellement-gunawan/
LOCATION:ENS Salle W
CATEGORIES:Variétés rationnelles
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