Image of pseudo-representations and coefficients of modular forms modulo p

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 = Sigma(infinity)(n=0) a(n)q(n) is such a form, for which we can assume without loss of generality that a(n) = 0 if (n, Np) > 1, calling delta(f) the density of the set of primes l such that a(l) not equal 0, we prove that delta(f) > 0 provided that f is not zero (and if p = 2, not a multiple of Delta). 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 delta(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. (C) 2019 Elsevier Inc. All rights reserved.