ICTP Mathematics Seminar (Victoria CANTORAL FARFAN) - Thursday, 9 November, at 14:00

[ICTP Math Section]

M A T H E M A T I C S   S E M I N A R S  2017


Thursday, 9 November, at 14:00 hrs.




Victoria Cantoral-Farfán (ICTP)


Title:
Torsion for abelian varieties of type III and new cases of the 
Mumford-Tate conjecture


Abstract:
Mordell-Weil's theorem states that, for an abelian variety defined over 
a number field K, the group of K-rational points is finitely generated. 
More precisely it can be seen as a product of a free group by a finite 
subgroup of torsion points over K. One can wonder if we can get a 
uniform bound for the order of the subgroup of torsion points over a 
finite extension L over K, depending on the degree of this extension and 
the dimension of the abelian variety, when the abelian variety varies in 
a certain class. This question is commonly known as the ``Strong Uniform 
Boundedness Conjecture''. For elliptic curves defined over a number 
field K, Merel proved in 1994 that we can indeed get a uniform bound 
using methods developed by Mazur and Kamienny.
A complementary question would be to ask if we can get a bound for the 
order of the subgroup of torsion points over a finite extension L over 
K, depending on the degree of this extension and the dimension of the 
abelian variety, when L varies over all the finite extensions of K and 
the abelian variety is fixed. This question had been already answered by 
Hindry and Ratazzi for certain classes of abelian varieties.
In this talk we focus our attention on this last question and extend the 
previous results. We are going to present some new results concerning 
the class of abelian varieties of type III in Albert's classification 
and “fully of Lefschetz type” (i.e. whose Mumford-Tate group is the 
group of symplectic or orthogonal similitudes commuting with 
endomorphisms and which satisfy the Mumford-Tate conjecture). Moreover, 
we are going to give an explicit list of abelian varieties which satisfy 
Mumford-Tate conjecture.




VENUE:  Stasi Seminar Room (ICTP Leonardo da Vinci Building, first floor)