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 Pk(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 Gouvêa and Mazur, we strengthen a congruence in [W] between coefficients of Pk and Pk′ for k ′ p-adically close 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 N0 of P0, which coincides with N0 infinitely often. As a consequence, we find a polygonal curve above N0. This tightest bound on N0 yields the strongest congruences between coefficients of P0 and Pk for k of large 3-adic valuation. 1 Overview and background Let p be a prime number, N be a positive integer relatively prime to p, and k be an integer. Let B be a p-adic ring between Zp and Øp, the ring of integers in Cp. Denote by Mk(N,B) the p-adic overconvergent modular forms of tame level N and weight k and by Sk(N,B) the subspace of overconvergent cusp forms. For every weight k, Atkin’s U operator is an endomorphism of Mk(N,B) stabilizing Sk(N,B). Denote by U (k) the restriction of U to Mk(N,B) and by U(k) the restriction to Sk(N,B). These are compact operators, so the characteristic series Pk(t) = det(1− tU ), Qk(t) = det(1− tU(k)) exist. Let am(Pk) be the coefficient of t m in Pk(t). As a function on a suitably defined space of weights k, am(Pk) is a rigid analytic function of k. Wan[W], and Buzzard[B] construct N̂(m), which grows as O(m2) and depends on p and N and not on k such that vp(am(Pk)) > N̂(m). Gouvêa and Mazur[GM] note, in an earlier work, the existence of N̂(m) and show, for prime p ≥ 5, integer l and positive integer n, vp(am(Pk)− am(Pk+lpn(p−1))) ≥ n+ 1. (1)