Alladi Ramakrishnan Hall

Serre’s problem on Galois action on points of division of elliptic curves

L. Merel

University of Paris VII

For E (non CM) elliptic curve over a number field K and n integer > 0, the absolute
Galois group G of K acts on points of n-division of E. Denote by I(E,n) the image of
G in the group of automorphism of E[n]. Serre proved in 1971 that the index of
I(E,n) is bounded in terms of E and K when n varies. He asked whether the bound can
be made independent of E (at least when K is the field of rational number). We will
explain how this problem is decomposed in subcases and what is the state of the art.
The latest advance is the theorem of Bilu-Parent (completed by Bilu-Parent-Rebolledo
in 2011).

Prerequisite : Familiarity with elliptic curves. Much of the talk will be about
group theory of GL_2(F_p), where F_p is the field with p elements.