Alexandru Buium (New Mexico)
Independence of modular points on elliptic curves
Abstract: We present global and local results about linear independence of CM points
on modular elliptic curves. The main global result (proved using equidistribution)
states that given a correspondence between a modular curve and an elliptic curve A,
the intersection of any finite rank subgroup of A
with the set of CM-points of A is finite. The local results
(proved via a method involving "arithmetic differential equations")
give quantitative versions of similar statements.
The latter method applies also to certain infinite rank subgroups,
and to the situation where the set of CM-points
is replaced by certain isogeny classes of points on the modular curve.
Finally, we prove Shimura curve analogues of these results.
All of this is joint work with B.Poonen.