[Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (07/12/2017, P. Allen, L. Goldmaker) ET COLLOQUE (VENDREDI)
Guillermo Martinez-Zalce
martinez at crm.umontreal.ca
Mon Dec 4 11:27:06 EST 2017
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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY
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DATE :
Le jeudi 7 décembre 2017 / Thursday, December 7, 2017
HEURE / TIME :
10 h 30 - 12 h / 10:30 a.m. - 12:00 p.m.
CONFERENCIER(S) / SPEAKER(S) :
Patrick Allen (UIUC)
TITRE / TITLE :
Automorphy of mod 3 representations over CM fields
LIEU / PLACE :
McGill University, Burnside Hall salle BH920
RESUME / ABSTRACT :
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses, as a starting point, the Langlands-Tunnell theorem, which implies that the mod 3 Galois representation attached to an elliptic curve over the rationals arises from a modular form of weight one. In order to feed this into modularity lifting theorems, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over CM fields, and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 representations over CM field arise from the "correct" automorphic forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. This is joint work with Chandrashekhar Khare and Jack Thorne.
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DATE :
Le jeudi 7 décembre 2017 / Thursday, December 7, 2017
HEURE / TIME :
14 h - 15 h 30 / 2:00 p.m. - 3:30 p.m.
CONFERENCIER(S) / SPEAKER(S) :
Leo Goldmaker (Williams College)
TITRE / TITLE :
Some refinements of Artin's conjecture
LIEU / PLACE :
Concordia University, Library Building, 9th floor, room LB 921-4
RESUME / ABSTRACT :
In 1927, Artin gave a heuristic argument that 2 is a primitive root (mod p) approximately 37% of the time. No one has been able to make his argument rigorous, and even the weaker problem of showing that 2 is a primitive root (mod p) for infinitely many p remain open.
Artin's initial heuristic has been generalized, giving rise to conjectures on the proportion of primes p for which any given integer is a primitive root (mod p); the general form of this is known as Artin's conjecture. In this talk I will describe several new conjectures (joint with Greg Martin, UBC) on the proportion of the time a given integer is "almost" a primitive root (mod p). Our conjectures subsume Artin's conjecture, and are borne out in computations. I'll also prove that our conjectures hold on average, and derive some consequences of this. For example, we obtain a new proof that Artin's conjecture holds on average (a result due to Goldfeld).
http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html <http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html>
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COLLOQUE DES SCIENCES MATHÉMATIQUES DU QUÉBEC
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DATE :
Le vendredi 8 décembre 2017 / Friday, December 8, 2017
HEURE / TIME :
16 h / 4:00 p.m.
CONFERENCIER(S) / SPEAKER(S) :
James Maynard (University of Oxford)
TITRE / TITLE :
Primes with missing digits
LIEU / PLACE :
UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, salle PK-5115
RESUME / ABSTRACT :
Many famous open questions about primes can be interpreted as questions about the digits of primes in a given base. We will talk about recent work showing there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most X^{1-c} elements less than X) which is typically very difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.
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Responsable(s) :
Olivier Collin (collin.olivier at uqam.ca)
Iosif Polterovich (iosif.polterovich at umontreal.ca)
Henri Darmon (darmon at math.mcgill.ca)
David A. Stephens (dstephens at math.mcgill.ca)
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http://www.crm.umontreal.ca/Colloques/colloqueSMQ-Montreal.html <http://www.crm.umontreal.ca/Colloques/colloqueSMQ-Montreal.html>
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