Gouvêa, Fernando Q. Where the slopes are. (English) Zbl 1047.11052 J. Ramanujan Math. Soc. 16, No. 1, 75-99 (2001). Fix a prime number \(p\). Assume \(p\nmid N\), and let \(f\in S_k(N_p,\mathbb{C}_p):= S_k(N,\mathbb{Q})\otimes \mathbb{C}_p\) be an eigenform for the Atkin-Lehner \(U\) operator: \(U(f)=\lambda f\). Let \(\text{slope}(f)= \text{ord}_p(\lambda)\). The author investigates numerically the distribution of the slopes of \(U\) for fixed level and varying weight. The computations suggests several interesting questions (conjectures): (i) the slopes are (much) smaller than expected (Questions 1, 2, 3, 4), (ii) the slopes are almost always integers (Question 5), (iii) one hopes for a representation-theoretic characterization of eigenforms that are of exceptional (see p. 85) or non-integral slope (Question 8). In the case of classical modular forms, J.-P. Serre [J. Am. Math. Soc. 10, 75–102 (1997; Zbl 0871.11032)] and J. B. Conrey, W. Duke and D. W. Farmer [Acta Arith. 78, 405–409 (1997; Zbl 0876.11020)] investigated the distribution as \(k\to\infty\) of all the eigenvalues of the \(p\)th Hecke operator \(T_p\) corresponding to eigenforms of weight \(k\). Reviewer: A. Dąbrowski (Szczecin) Cited in 1 ReviewCited in 9 Documents MSC: 11F85 \(p\)-adic theory, local fields 11F25 Hecke-Petersson operators, differential operators (one variable) Keywords:Atkin-Lehner operator; Hecke operator; \(p\)-adic modular form; slope; eigenvalue Citations:Zbl 0871.11032; Zbl 0876.11020 PDFBibTeX XMLCite \textit{F. Q. Gouvêa}, J. Ramanujan Math. Soc. 16, No. 1, 75--99 (2001; Zbl 1047.11052) Full Text: arXiv