P-adic Slope Distributions for Modular Forms

The Sato--Tate conjecture predicts statistical properties of point counts on elliptic curves. Its proof over the rationals relies on elliptic curves being modular. In the modular world, there is a wider class of distributions to investigate. For instance, Serre and Conrey--Duke--Farmer established (c. 2000) the distribution of fixed Hecke eigenvalues in modular weights growing to infinity.
This seminar will survey non-Archimedean analogues of these distributions. We first describe the p-adic slope problem for modular eigenforms. Then, we survey the past 20-30 years of research on slopes. One aim is Liu, Truong, Xiao, and Zhao's recent results on the "ghost conjecture" of myself and Pollack. Among its by-products, we learn about p-adic distributions in weight aspects, as previously predicted by Gouvêa.