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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.

11F33 Congruences for modular and \(p\)-adic modular forms
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