### Bounding the torsion in CM elliptic curves, by Dipendra Prasad and C.S. Yogananda

Merel has shown that the order of torsion subgroup of an elliptic
curve over a number field can be bounded in terms of only the
degree of the number field. The purpose of this note is to investigate
what could be the `right bound'.
In this paper we use the result of Deuring on the supersingular
primes for a CM elliptic curves to give estimate on the number
of torsion points on CM elliptic curves over any number field
which depends only linearly with the degree of the number field.
This result is also a simple corollary of a result of
A. Silverberg bounding the order of torsion elements which was
proved using the main theorem of
Complex multiplication. We also give a heuristic for hoping that the
number of torsion points on any elliptic curve be a bounded linearly
with the degree of the number field.

Dipendra Prasad and C.S. Yogananda <dprasad@mri.ernet.in, yoga@math.imsc.ernet.in>