  We study the relationship between recent conjectures on slopes of
overconvergent p-adic modular forms "near the boundary" of p-adic weight space.
We also prove in tame level 1 that the coefficients of the Fredholm series of
the U_p operator never vanish modulo p, a phenomenon that fails at higher
level. In higher level, we do check that infinitely many coefficients are
non-zero modulo p using a modular interpretation of the mod p reduction of the
Fredholm series recently discovered by Andreatta, Iovita and Pilloni.
