[Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (01/12/2016, Dinakar Ramadrishnan)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Mon Nov 28 11:17:59 EST 2016


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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 1 décembre 2016 / Thursday, December 1, 2016

HEURE / TIME :
10 h 30 - 12 h / 10:30 a.m. - 12:00 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Dinakar Ramadrishnan (Caltech)

TITRE / TITLE :
Rational Points on Picard Modular Surfaces

LIEU / PLACE :
McGill University, Burnside Hall salle BH920

RESUME / ABSTRACT :
A basic question in Number theory, going back to Diophantos, is to understand the rational solutions of systems of polynomial equations with integer coefficients. This is recast usually in terms of rational points on the algebraic variety V defined by the polynomial system. A very intriguing principle is that the topology and geometry of the complex points of V have a strong bearing on the set V(Q) of rational points. For example, when V is a smooth projective curve, i.e., when V(C) is a compact Riemann surface, the genus g of V has the following impact: When g=0 there are many rational points or none, while for g=1, the cardinality of V(Q) can be finite or infinite. A striking theorm of Faltings proved that V(Q) is always finite when g is >1; in this case, X(C) is uniformized by the hyperbolic plane. For algebraic surfaces V, a beautiful conjecture of Lang asserts such a finiteness result when its is hyperbolic, in the sense that there is no non-constant map from C to V(C). We will discuss this conjecture and discuss what happens in the special case when V(C) is a Picard modular surface X, which arises classically as a (compactification of) a quotient of the unit ball in C^2 by an arithmetic lattice Gamma in SU(2,1). Such surfaces have mirific properties, making them a crucial place to test various arithmetic and geometric conjectures. 
This talk will begin by describing the albanese variety and its residual quotients, and then Lang's conjecture over finitely generate fields. It will then move on to discuss an ongoing, partially completed, program with M. Dimitrov to establish the paucity of rational points on the open part. We will also discuss the connection to the problem of uniform boundedness of torsion for abelian varieties of dimension <4 with multiplication by an imaginary quadratic field.


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DATE :
Le jeudi 1 décembre 2016 / Thursday, December 1, 2016

HEURE / TIME :
14 h - 15 h 30 / 2:00 p.m. - 3:30 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Julie Desjardins (Université Paris VII)

TITRE / TITLE :
Densité des points rationnels sur les surfaces elliptiques / Density of rational points on elliptic surfaces

LIEU / PLACE :
Concordia Bookstore, Library Building, 9th floor

RESUME / ABSTRACT :
Densité des points rationnels sur les surfaces elliptiques

Resume: Pour une surface algébrique X, on s'intéresse à l'ensemble des points rationnels X(ℚ). Est-il non-vide ? Est-il infini ? Est-il dense pour la topologie de Zariski ? Nous nous intéresserons aux surfaces elliptiques, les familles à un paramètre de courbes elliptiques. La densité des points rationnels de ce type de surfaces est encore mal connue en général. Lorsque la surface est isotriviale, on démontre la densité dans la plupart des cas lorsqu'elle est rationnelle. De plus, en étudiant la variation du signe de l'équation fonctionnelle des fibres on peut prédire la densité pour les surfaces non isotriviales conditionnellement à plusieurs conjectures (conjecture de parité, squarefree conjecture et conjecture de Chowla). Ces conjectures imposent une restriction sur le degré des facteurs irréductibles du discriminant. On peut les éviter dans certains cas, et ainsi obtenir la variation du signe inconditionnelle sur plusieurs familles de surfaces elliptiques dont le discriminant a des facteurs de degré arbitrairement grand.

Title: Density of rational points on elliptic surfaces

Abstract: Let X be an algebraic surface. We are interested in the set of rational points X(ℚ). Is it non-empty ? Is it infinite ? Is it Zariski-dense ? We are concerned with elliptic surfaces, i.e. 1-parameter families of elliptic curves. The density of rational points is not well known in general. When the surface is isotrivial, one shows the density in most cases when it is also rational. Moreover, by studying the variation of the root number of the fibers, one predicts the density on non-isotrivial elliptic surfaces conditionally to some conjectures (parity conjecture, squarefree conjecture, Chowla's conjecture). The last two conjectures impose a restriction on the degree of the factors of the discriminant. We manage to avoid the squarefree conjecture in certain cases, and thus show unconditionally the variation of the root number, without imposing a bound for the degree of the irreducible factors.


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Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)
Eyal Z. Goren (eyal.goren at mcgill.ca)
Chantal David (cdavid at mathstat.concordia.ca)
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http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html <http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html>


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2016 Montreal-Toronto Workshop in Number Theory

December 8-9, 2016 at the CRM

http://www.crm.umontreal.ca/2016/MTWNT2016/index_e.php <http://www.crm.umontreal.ca/2016/MTWNT2016/index_e.php>

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