Arithmetic properties of Fredholm series for 𝑝-adic modular forms

Arithmetic properties of Fredholm series for 𝑝-adic modular forms Bergdall, John; Pollack, Robert 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 coeffcients of the Fredholm series of the U𝑝 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.