[Liste-CICMA] CRM-Fields 2016 Montreal-Toronto workshop, Fields Institute, May 12-13, 2016

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Mon Feb 22 09:14:48 EST 2016

 The joint CRM-Fields 2016 Montreal-Toronto workshop on \Serre's uniform boundedness con-
jecture" will take place a the Fields Institute on May 12-13, 2016.
Let E  be an elliptic curve over Q  without complex multiplication, and let `  be a prime. Serre
showed in 1972 that the image of the map from the absolute Galois group GalQ  = Gal(Q=Q ) to
the group of automorphisms Aut(Etors ) '  GL2 (^Z ) is open, and then of nite index in GL2 (^Z ). In
particular, it follows that there exists a constant c (E ) depending on E  such that E;` (GalQ ), the
image of the Galois representation given by the action of the absolute Galois group on the ` -torsion
points of E, is surjective in GL2 (Z=`Z ) when ` > c (E ). One of the main tasks of Serre's paper is to
show that such a c (E ) exists. Serre also asked if the constant c (E ) can be bounded independently of
E , and moreover do we always have c (E ) <  37 for non-CM elliptic curves. This came to be known
as \Serre's uniform boundedness conjecture". If E;`  is not surjective, then its image is contained
in a maximal subgroup M  of GL2 (Z=`Z ). If this maximal subgroup is a Borel subgroup, this can
only happen for `   17 or `  = 37 (this is a famous result of Mazur). Thus, two cases remain to be
treated to prove Serre's uniform boundedness conjecture: when M  is a normalizer of a split Cartan
subgroup, or when M is the normaliser of a non-split Cartan subgroup. The rst case was recently
proved in a breakthrough work of Bilu and Parent (2011), and they show that there exists a uniform
bound `0  such that E;`  is not contained in the normalizer of a split Cartan subgroup for all `  `0 ,
and all non-CM elliptic curves over Q . Moreover, one can take `0  = 13.
Understanding the argument of Bilu and Parent will be the core of the workshop. The main tool
is the study of the moduli spaces Ysplit (` ), and the statement can be rephrased as saying that the
only rational points on the curves Ysplit (` ) are CM points when ` > `0 . This is done by using Siegel
functions, Falting heights and a crucial improvement of a classical theorem of Masser and Wustholz
about Q -isogenies of bounded degree.
One can also ask what are the best known bounds for c (E ) (including then the non-split Car-
tan case). There is a vast literature on the subject, including Serre's original bound c (E )
c  logN (log log 2N )3  under the GRH (where N  is the conductor of E ) and the bound of Masser
and Wustholtz in terms of the height of the j -invariant of the curves. Recently, Zywina also gave
explicit bounds for the index [GL2 (Z ) : E (GalQ )], which is a more general question than bounding
c (E ). We are also interested in analytic applications of Serre's open image theorem and Serre's
uniform boundedness conjecture.
One of the main goals of the workshop is to foster scientic interaction between the number
theory research groups in Montreal and Toronto, and nearby universities in Canada and the United
States, at the whole range of seniority, from professors to beginning graduate students. The talks
will consist of a mixture of presentations by graduate students presenting the background on Galois
representations attached to elliptic curves and the classical work of Serre, of presentations of more
specialized material by some faculty members, and of general overview talks about current research
done by members of both number theory groups.

List of speakers (tentative and to be completed):

  Jake Chinis (Concordia University), Abishek Oswal (University of Toronto) and Valentine
Chiche-Lapierre (University of Toronto): Background on `-adic representations and Serre's
open image theorem.
Jacob Tsimerman (University of Toronto): Runge's theorem and the work of Bilu-Parent.

  Jan Vonk (McGill University): The cursed modular curve.

  David Zywina (Cornell University): Bounds for Serre's open image theorem.

  Chantal David (Concordia University): Analytic applications of bounds for Serre's open
image theorem.

  Dimitris Koukoulopoulos (Universite de Montreal): Statistics for elliptic curves and random
Euler products.

Guillermo Martinez-Zalce
Responsable des laboratoires
Centre de recherches mathématiques
Université de Montréal
Case postale 6128, Succursale centre-ville
Montréal (Québec)
H3C 3J7
tel: (514) 343-7574
      (514) 398-3826 (mercredis)

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