Author's Final Manuscript
Proceedings of the London Mathematical Society
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 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.
Bergdall, John and Robert Pollack. 2016. "Arithmetic properties of Fredholm series for p-adic modular forms." Proceedings of the London Mathematical Society, 113.3: 419-444.