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Where the slopes are. (English) Zbl 1047.11052

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 \(k\).

MSC:

11F85 \(p\)-adic theory, local fields
11F25 Hecke-Petersson operators, differential operators (one variable)
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