[Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (23/02/2017, Brian Hwang, Taylor Dupuy)
Guillermo Martinez-Zalce
martinez at crm.umontreal.ca
Tue Feb 21 08:50:48 EST 2017
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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY
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DATE :
Le jeudi 23 février 2017 / Thursday, February 23, 2017
HEURE / TIME :
10 h 30 - 12 h / 10:30 a.m. - 12:00 p.m.
CONFERENCIER(S) / SPEAKER(S) :
Brian Hwang (Cornell)
TITRE / TITLE :
Galois theory, non-abelian class field theory, and harmonic families of automorphic forms
LIEU / PLACE :
McGill University, Burnside Hall salle BH920
RESUME / ABSTRACT :
A number of questions in Galois theory can be phrased as follows: how "large" (in various senses) can the Galois group G of an extension of the rational numbers be, if the extension is only allowed to ramify at a "small" set of primes? If we assume that G is abelian, class field theory gives us a complete answer, but the question is open in almost every nonabelian case, because there is no known way to systematically and explicitly construct such extensions in full generality.
However, it turns out that if we shift our perspective slightly, we find a point where the problem's defense is weakest. While the question above is natural and the objects are familiar, we will see that to answer certain questions about the “largeness” of this Galois group, it seems necessary to use techniques involving automorphic forms and their representation-theoretic avatars. In particular, it will turn out that some recent results on “harmonic” families of automorphic forms (a notion we will explain) translate to the fact that such number fields, despite not being explicitly constructible by known methods, turn out to "exist in abundance" and allow us to find bounds on the sizes of such Galois groups.
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DATE :
Le jeudi 23 février 2017 / Thursday, February 23, 2017
HEURE / TIME :
14 h - 15 h 30 / 2:00 p.m. - 3:30 p.m.
CONFERENCIER(S) / SPEAKER(S) :
Taylor Dupuy (U. Vermont, Burlington)
TITRE / TITLE :
The Wittfinitesimal Torelli Problem
LIEU / PLACE :
Concordia University, Library Building, 9th floor, room LB 921-4
RESUME / ABSTRACT :
Kodaira-Spencer classes classify local deformations of schemes. In this talk we construct cohomology classes associated to smooth schemes over p-adic rings which behave like Kodaira-Spencer classes. These cohomology classes are obstructions to lifts of the Frobenius modulo p^2. We will explain how to construct these classes, explain how these classes behave under morphisms of varieties, and relate them to a problem of Coleman's concerning the non-existence of non-hyperelliptic CM Jacobians for large genus (Coleman's original conjecture was false). Whether or not Jacobians have CM turns out to be determined by the behavior of a "wittfinitesimal Torelli map". In the Kodaira-Spencer setting this map is called the "infinitesimal Torelli map" and is known to be injective outside the hyperelliptic locus (in characteristic zero).
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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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