Arithmetic properties of Fredholm series for \(p\)-adic modular forms (Q2834910)

The paper is concerned with a conjecture from a recent preprint of \textit{D. Wan}, \textit{L. Xiao} and \textit{J. Zhang} [``Slopes of eigencurves over boundary disks'', Preprint, \url{arXiv:1407.0279}], which, quoting the authors, makes precise a folklore possibility, regarding the slopes of \(p\)-adic modular forms near the ``boundary of weight space''. The conjecture comes in two parts. The first part says that the slopes of overconvergent \(p\)-adic eigenforms at weights approaching the boundary should change linearly with respect to the valuation of the weight. The second part of the conjecture is that these slopes, after normalizing by this conjectured linear change, form a finite union of arithmetic progressions. The main goal in this paper is to prove that the second conjecture is a consequence of the first.NEWLINENEWLINEImplicit in the conjecture is a simple and beautiful description of these slopes: namely near the boundary, the slopes arise from a scaling of the Newton polygon of the mod \(p\) reduction of the Fredholm series of the \(U_p\) operator. The first half of this paper is devoted to studying this characteristic \(p\) object using a \(p\)-adic version of the trace formula discovered by \textit{M. Koike} in [Nagoya Math. J. 56, 45--52 (1975; Zbl 0301.10026)]. The authors prove that this mod \(p\) Fredholm series is not a polynomial; that is, it is a true power series with infinitely many non-zero coefficients. This observation is an important step in the deduction of the main theorem.