## 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

### MSC:

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

Zbl 0773.11030