Gouvêa–Mazur conjecture on slopes of overconvergent modular forms

Let $p$ be a prime, $N$ a positive integer relatively prime to $p$, and $k$ an integer. Let ${\mathcal M}_k(N,B)$ denote the space of $p$-adic overconvergent modular forms of tame level $N$ and weight $k$, and let $U^{(k)}$ be the Atkin operator on this space. For rational $lpha$, let $d(k,\alpha)$ be the number of $U^{(k)}$ eigenvalues with $p$-adic valuation $\alpha$.

Gouvêa–Mazur conjecture. Let $k$, $l$ be integers, $n$ be a positive integer, and $\alpha<n$. Then

$$\nd(k,\alpha)=d\bigl(k+lp^n(p-1),\alpha\bigr).$$

The conjecture predicts local constancy of the multiplicities of low-slope $p$-adic eigenvalues as the weight varies in suitable $p$-adic congruence classes. The paper proves stronger coefficient congruences and explains that these imply the conclusion for an explicit quadratic range of slopes, but the full conjecture is left unresolved here.