Linear Independence of Hecke Operators in the Homology of X 0 ( N )

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 T 1 , T 2 , …, T D are linearly independent in their actions on the cycle e from 0 to i ∞ in H 1 ( X 0 ( N ) ( C ), Q ). In particular, he shows independence when max( D 8 , 400 D 4 ) < 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.