In Merel's recent proof [7] of the uniform boundedness conjecture for the torsion
of elliptic curves over number fields, a key step is to show that for sufficiently large
primes N, the Hecke operators T1, T2, …,
TD are linearly independent in their actions on the cycle e from 0 to
i∞ in H1(X0(N) (C), Q).
In particular, he shows independence when
max(D8, 400D4) < N/(log N)4.
In this paper we use analytic techniques to show
that one can choose D considerably larger than this, provided that N is large.