Bounding slopes of $p$-adic modular forms
Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and weight $k$. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's congruence between coefficients of $P_k$ and $P_{k'}$ for $k'$ close $p$-adically to $k$. For $p-1 | 12$, $N = 1$, $k = 0$, we compute a matrix for $U$ whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for $p = 3$ a parabola below the Newton polygon $N_0$ of $P_0$, which coincides with $N_0$ infinitely often. As a consequence, we find a polygonal curve above $N_0$. This tightest bound on $N_0$ yields the strongest congruences between coefficients of $P_0$ and $P_k$ for $k$ of large 3-adic valuation.