Joint IAS/Princeton University Number Theory Seminar

I will explain two recent results concerning non-zero coefficients of modular forms modulo a prime $p$. The first result, a joint work with K. Soundararajan, gives an asymptotic equivalent for the number of such coefficients. The second is concerned with the set of primes which are indices of non-zero coefficients. Such sets are Frobenian, hence have a natural density, and the result states that these densities are bounded below by a positive constant and above by a constant less than one, for all modular forms (of a fixed level) except for a few well-understood exceptions. I will briefly discuss the proof of this result, which is based on a study of the image of big Galois representations, and applications.