Continuity properties of \(p\)-adic modular forms. (English) Zbl 0829.11026

Kisilevsky, Hershy (ed.) et al., Elliptic curves and related topics. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 4, 85-100 (1994).
Fix a prime \(p\). Let \(D(k,\alpha)\) (resp. \(d(k, \alpha))\) denote the dimension of the space of classical modular forms of weight \(k\) on \(\Gamma_0(Np)\) with slope \(\alpha\) (resp. of \(p\)-adic modular forms of weight \(k\) with slope \(\alpha)\). The author considers the following Conjecture 1. Fix a slope \(\alpha\), and let \(k_1\), \(k_2\), be integers satisfying (i) \(k_1\), \(k_2 \geq 2 \alpha + 2\); (ii) \(k_1 \equiv k_2 \pmod {p^n(p - 1)}\) for some integer \(n \geq \alpha\). Then \(D(k_1, \alpha) = D(k_2, \alpha)\).
The case \(\alpha = 0\) is known due to Hida. The numerical evidence in the case \(\alpha \neq 0\) is given in a paper of F. Q. Gouvêa and B. Mazur, Families of modular eigenforms [Math. Comput. 58, 793-805 (1992; Zbl 0773.11030)]. The conjecture is reduced to the following two conjectures on \(p\)-adic modular forms.
Conjecture 2. Let \(k_1\) and \(k_2\) be integers such that \(k_1 \equiv k_2 \pmod {p^n (p - 1)}\). Then \(d(k_1, \alpha) = d(k_2, \alpha)\) for \(n \geq \alpha\).
Conjecture 3. Any overconvergent \(p\)-adic eigenform of slope \(\alpha\) and integral weight greater than or equal to \(2 \alpha + 2\) is a classical modular form.
For the entire collection see [Zbl 0788.00052].
Reviewer: A.Dabrowski


11F33 Congruences for modular and \(p\)-adic modular forms
14G20 Local ground fields in algebraic geometry


Zbl 0773.11030