On the independence of the Fourier coefficients of two p-adic modular forms

Abstract

We study rigidity phenomena for p-adic modular forms and their families. Extending a result of Coleman, we show that if two overconvergent modular forms have Fourier coefficients with values in a finite ratio set, then they must be scalar multiples of each other. We further prove that this rigidity persists for p-adic families varying over weight space.