[Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (15/12/2016, Djordjo Milovic, Lilian Pierce, Nuno Freitas)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Tue Dec 13 15:19:58 EST 2016

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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 15 décembre 2016 / Thursday, December 15, 2016

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

CONFERENCIER(S) / SPEAKER(S) :
Djordjo Milovic (IAS)

TITRE / TITLE :
On the 16-rank of class groups

LIEU / PLACE :
McGill University, Burnside Hall salle BH920

RESUME / ABSTRACT :
The study of class groups of quadratic number fields goes back to Gauss, who proved that the 2-rank (i.e., the "width" of the 2-part) of the (narrow) class group of the quadratic number field of discriminant D is equal to $\omega(D)-1$, where $\omega(D)$ is the number of distinct prime divisors of D. If D is of the form -8p, with p a prime number, then the 2-part of the class group is cyclic and hence determined by the largest power of 2 dividing the class number. Simple Chebotarev-type criteria for divisibility by 4 and by 8 were found by Redei and Reichardt in the 1930's. We prove that the natural density of the set of prime numbers p congruent to -1 modulo 4 such that 16 divides the class number is equal to 1/16. Our proof suggests that there is no simple Chebetarev-type criterion for divisibility by 16. If time allows, we will also discuss the 16-rank in other families of discriminants and related analytic problems.

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DATE :
Le jeudi 15 décembre 2016 / Thursday, December 15, 2016

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

CONFERENCIER(S) / SPEAKER(S) :
Lillian Pierce (Duke University)

TITRE / TITLE :
p-torsion in class groups of number fields of arbitrary degree

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

RESUME / ABSTRACT :
Fix a number field K of degree n over the rationals, and a prime p, and consider the p-torsion subgroup of the class group of K. How big is it? It is conjectured that this p-torsion subgroup should be very small (in an appropriate sense), relative to the absolute discriminant of the field. But it has so far proved extremely difficult even to beat the trivial bound, that is, to show that the p-torsion subgroup is noticeably smaller than the full class group. In 2007, Ellenberg and Venkatesh shaved a power off the trivial bound by assuming GRH. This talk will discuss two contrasting methods that recover this bound for certain families of fields, without assuming GRH. This includes recent joint work with Jordan Ellenberg, Melanie Matchett Wood, and Caroline Turnage-Butterbaugh.

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DATE :
Le jeudi 15 décembre 2016 / Thursday, December 15, 2016

HEURE / TIME :
16 h - 17 h 30 / 4:00 p.m. - 5:30 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Nuno Freitas (UBC)

TITRE / TITLE :
Galois representations and the Generalized Fermat Equation.

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

RESUME / ABSTRACT :
Wiles' proof of Fermat's Last Theorem gave birth to the 'modular method' to attack Diophantine equations. Since then many other equations were solved using generalizations of this method. However, the success of the generalizations relies on a final "contradiction step" which is invisible in the original proof.

In this talk, we will recall the modular method and discuss this contradiction step. We will discuss why developing methods to distinguish Galois representations is relevant to it  and, in particular, explain how "the symplectic argument" sometimes
allows to overcome this last step. We will illustrate these tools with example of applications to special cases of the Generalized Fermat equation x^r + y^q = z^p.

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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