# [Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (12/10/2017, Udi De Shalit, Corentin Perret-Gentil)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Tue Oct 10 09:51:48 EDT 2017

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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 12 octobre 2017 / Thursday, October 12, 2017

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

CONFERENCIER(S) / SPEAKER(S) :
Udi De Shalit (The Hebrew University, Jerusalem)

TITRE / TITLE :
Foliations on unitary Shimura varieties in positive characteristic.

LIEU / PLACE :
McGill University, Burnside Hall salle BH920

RESUME / ABSTRACT :
Let E be a quadratic imaginary field and p a prime which is inert in E. Let S  be the special fiber (at p) of a unitary Shimura variety of signature (n,m) and hyperspecial level subgroup at p, associated with E/Q.

We study a natural foliation in the tangent bundle of S, which is originally  defined on the \mu-ordinary stratum only, but is extended to a certain non-singular blow-up of S. We identify the quotient of S by the foliation with a certain irreducible component of a Shimura variety with parahoric level structure at p. As a result we get new results on the singularities of the latter.

We study integral submanifolds of the foliation and end the talk with a new conjecture of Andre-Oort type.

This is joint work with Eyal Goren (McGill).

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DATE :
Le jeudi 12 octobre 2017 / Thursday, October 12, 2017

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

CONFERENCIER(S) / SPEAKER(S) :
Corentin Perret-Gentil (CRM)

TITRE / TITLE :
Reduced exponential sums, their monodromy groups, and their distribution

LIEU / PLACE :
Concordia University, Library Building, 9th floor, room LB 921-4

RESUME / ABSTRACT :
The Deligne-Katz equidistribution theorem asserts that hyper-Kloosterman sums over $\mathbb{F}_q$ of fixed degree equidistribute in a compact subset of the complex numbers, with respect to the pushforward of a Haar measure, as $q\to+\infty$. A interesting variant is to view general exponential sums as lying in the cyclotomic integers $\Z[\zeta_p]$, and ask similar questions about their distribution, or that of their reductions in residue fields. In this talk, we will investigate these questions for Kloosterman sums. In the first part, we will show how the theory of $\ell$-adic trace functions over finite fields can be used for that matter, once the monodromy groups are known. In the second part, we will discuss the determination of integral monodromy groups of Kloosterman sheaves. Finally, we will discuss more specific applications.

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Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)
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