Abstract
We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level N modulo a prime p, we prove new results about the coefficients of modular forms mod p. If f=∑n=0∞anqn is such a form, for which we can assume without loss of generality that an=0 if (n,Np)>1, calling δ(f) the density of the set of primes ℓ such that aℓ≠0, we prove that δ(f)>0 provided that f is not zero (and if p=2, not a multiple of Δ). More importantly, we prove, when p>2, a uniform version of this result, namely that there exists a constant c>0 depending only on N and p such that δ(f)>c for all forms f except for those in an explicit subspace of infinite codimension of the space of all modular forms mod p of level N. Forms in this subspace, called special modular forms mod p, are proved to be closely related to certain classes of modular forms mod p previously studied by the author, Nicolas and Serre, called cyclotomic and CM modular forms mod p.