[Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (14/12/2017, Bryden Cais, Anna Haensch)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Mon Dec 11 13:51:43 EST 2017


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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 14 décembre 2017 / Thursday, December 14, 2017

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

CONFERENCIER(S) / SPEAKER(S) :
Bryden Cais (University of Arizona)

TITRE / TITLE :
A Motivic Deuring Shafarevich Formula

LIEU / PLACE :
McGill University, Burnside Hall salle BH920

RESUME / ABSTRACT :
Let Y --> X be a branched G-covering of curves over a field k.  The genus of X and the genus of Y are related by the famous Hurwitz genus formula.  
When k is perfect of characteristic p and G is a p-group, one also has the Deuring-Shafarevich formula which relates the p-rank of X to that of Y.  
In this talk, we will discuss our attempts to find a "motivic" generalization of the Deuring Shafarevich formula by studying how the p-torsion group schemes 
of the Jacobians of X and Y are related.  In particular, we will explain how to promote the numerical Deuring--Shafarevich formula to an isomorphism of (etale) group schemes. This is ongoing joint work with Rachel Pries.

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DATE :
Le jeudi 14 décembre 2017 / Thursday, December 14, 2017

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

CONFERENCIER(S) / SPEAKER(S) :
Anna Haensch (Duquesne University)

TITRE / TITLE :
Bridging the algebraic and analytic theory of the representation problem for quadratic polynomials

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

RESUME / ABSTRACT :
Given a polynomial f(x) of several variables with rational coefficients and an integer n, we say that f represents n if the equation f(x)=n is solvable in the integers.  One might ask, is it possible to effectively determine the set of integers represented by f? This so-called representation problem for quadratic polynomials is one of the classical problems in number theory.  The negative answer to Hilbert's 10th problem tells us that in general, there is no finite algorithm to decide whether a solution exists.  However work of Siegel in the 1970’s shows that in the case of quadratic polynomials the representation problem is tractable.  Algebraically, we can view the representation problem using the language of integral quadratic lattices and lattice cosets, but this method stops short of realizing a full local-global principle.  Melding this with the analytic approach of studying the coefficients of theta series we can reach some satisfying solutions to the representation problem for certain families of polynomials.  In this talk we will illuminate several important connections between the algebraic and analytic theory of quadratic lattice and finish with a conjecture involving a Siegel-Well type formula for representations by inhomogeneous quadratic polynomials.

Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)
Adrian Iovita (adrian.iovita at concordia.ca)
Maksym Radziwill (maksym.radziwill at gmail.com)

http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html <http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html>



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