Images of Pseudo-Representations and Coefficients of Modular Forms modulo p

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 $f=\sum_{n=0}^\infty a_n q^n$ 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 $\ell$ such that $a_\ell \neq 0$, 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 {\it 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 {\it 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 $p$.