Let $E_k$ be the normalized Eisenstein series of weight k and let p be an odd prime. Then
$$ E_{p^m(p-1)} = 1 \mod p^{m+1}, $$
and so the p-adic limit $\lim E_{p^m(p-1)} = 1$ is a p-adic modular form of weight 0. (It is even overconvergent.)
Question: Suppose f is a p-adic modular form whose q-expansion is a polynomial. Is the q-expansion of f a constant?
It is. I want to argue the following way: if the polynomial is non-constant then after scaling it has integral coefficients and so the reduction of the p-adic form mod p^n will be a classical form whose q-expansion is a non-constant polynomial. But I think Katz proved in his Antwerp paper that the only modular forms which are polynomials in q are the constants, which is a contradiction. One needs to dot some i's and cross some t's here, but I am optimistic.
