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Dimension variation of classical and $$p$$-adic modular forms. (English) Zbl 0907.11016
Let $$p\geq 5$$ be a fixed prime and let $$N$$ be a fixed positive integer not divisible by $$p$$. Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$. For a nonnegative integer $$k$$, denote by $$M_k(Np,K)$$ the space of classical modular forms of weight $$k$$ on $$\Gamma_1 (N) \cap \Gamma_0(p)$$ whose Fourier coefficients belong to $$K$$. We have the direct sum decomposition $M_k(Np,K) =\oplus_{\alpha\in \mathbb{Q}_{>0}} M_k(Np, K)^{(\alpha)},$ under the action of the Atkin $$U$$-operator, where each root of the characteristic polynomial of $$U$$ on $$M_k (Np,K)^{(\alpha)}$$ has $$p$$-adic valuation $$\alpha$$.
Let $$d(k,\alpha): =\dim M_k (Np,K)^{ (\alpha)}$$. Let $$m(\alpha)$$ be the smallest non-negative integer (or $$+\infty$$ if it does not exist) such that whenever $$k_1$$ and $$k_2$$ are two positive integers $$\geq 2\alpha+2$$ satisfying $$k_1 \equiv k_2 \pmod {p^{m (\alpha)} (p-1)}$$, then we have the equality $$d(k_1, \alpha)= d(k_2, \alpha)$$. Conjecture (Gouvêa-Mazur): $$m(\alpha) \leq[\alpha]$$.
The author gives a quadratic upper bound for $$m(\alpha)$$ (Theorem 1.1). The proof is based on Katz’s theory of overconvergent $$p$$-adic modular forms, and recent work of Coleman and Gouvêa-Mazur.

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms
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