Divisibility of coefficients of modular forms
Divisibility of coefficients of modular forms

Joel Bellaiche , Brandeis University
Fine Hall 214
I will explain two recent results concerning nonzero 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 nonzero 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 wellunderstood 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.