Bergdall, JohnPollack, Robert2019-03-132019-03-132016J Bergdall, R Pollack. 2016. "Arithmetic properties of Fredholm series for p-adic modular forms." Proceedings of the London Mathematical Society, Volume 113, Issue 4, pp. 419 - 444.0024-6115https://hdl.handle.net/2144/34274We 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.p. 419 - 444en-USPure mathematicsArticle