“…When the index m = 1, the question about ker D 0 , which is nothing but the restriction map from J k,m (N) to M k (N), the space of elliptic modular forms of weight k on Γ 0 (N); defined by φ(τ, z) → φ(τ, 0), translates into the possibility of removing the differential operator D 2 while preserving injectivity. This question is also interesting in its own right, and has received some attention in the recent past, see the works [1,2,3,9], and the introduction there. We only note here that first results along this line of investigation seems to be by J. Kramer, who gave an explicit description of ker D 0 when m = 1 and level N a prime, in terms of the vanishing order of cusp forms in a certain subspace of S 4 (N) (This is related to the so-called Weierstrass subspaces of S k (N), see [2]).…”