Slopes of eigencurves over boundary disks

Let $p$ be a prime number. We study the slopes of $U_p$-eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra. We give a sharp lower bound of the corresponding Newton polygon. The computation happens over a definite quaternion algebra by Jacquet-Langlands correspondence; it generalizes a prior work of Daniel Jacobs who treated the case of $p=3$ with a particular level. In case when the modular forms have a finite character of conductor highly divisible by $p$, we improve the lower bound to show that the slopes of $U_p$-eigenvalues grow roughly like arithmetic progressions as the weight $k$ increases. This is the first very positive evidence for Buzzard-Kilford's conjecture on the behavior of the eigencurve near the boundary of the weight space, that is proved for arbitrary $p$ and general level. We give the exact formula of a fraction of the slope sequence.