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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 Jacobs (Slopes of compact hecke operators. Thesis, University of London, Imperial College, 2004) 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.

Slopes of eigencurves over boundary disks