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Classical and overconvergent modular forms. (English) Zbl 1073.11515

This is the companion paper to Invent. Math. 124, No. 1-3, 215–241 (1996; Zbl 0851.11030).
From the introduction: The purpose of this article is to use rigid analysis to clarify the relation between classical modular forms and Katz’s overconvergent forms. In particular, the author proves a conjecture of F. Gouvêa [CRM Proc. Lect. Notes 4, 85–99 (1994; Zbl 0829.11026) Conj. 3] which asserts that every overconvergent \(p\)-adic modular form of sufficiently small slope is classical.
More precisely, let \(p>3\) be a prime, \(K\) a complete subfield of \(\mathbb{C}_p\), \(N\) be a positive integer such that \((N,p)=1\) and \(k\) an integer. Katz has defined the space \(M_k (\Gamma_1(N))\) of overconvergent \(p\)-adic modular forms of level \(\Gamma_1(N)\) and weight \(k\) over \(K\) and there is a natural map from weight \(k\) modular forms of level \(\Gamma_1(Np)\) with trivial character at \(p\) to \(M_k(\Gamma_1(N))\). These modular forms are called classical modular forms. In addition, there is an operator \(U\) on these forms such that if \(F\) is an overconvergent modular form with \(q\)-expansion \(F(q)=\sum_{n\geq 0}a_nq^n\) then \[ UF(q)=\sum_{n\geq 0} a_{pn}q^n. \] Then the author proves (Theorem 6.1), that if \(F\) is a generalized eigenvector for \(U\) with eigenvalue \(\lambda\) (i.e., in the kernel of \((U-\lambda)^n\) for some positive integer \(n)\) of weight \(k\) and \(\lambda\) has \(p\)-adic valuation strictly less than \(k-1\), then \(F\) is a classical modular form. In this case the valuation of \(\lambda\) is called the slope of \(F\).
In the case when \(F\) has slope 0, this is a theorem of H. Hida [Ann. Sci. Ec. Norm. Supér. (4) 19, 231–273 (1986; Zbl 0607.10022)] and, more generally, it implies Gouvêa’s conjecture mentioned above (which is the above conclusion under the additional hypothesis that the slope of \(F\) is at most \((k-2)/2)\). This almost settles the question of which overconvergent eigenforms are classical, as the slope of any classical modular form of weight \(k\) is at most \(k-1\).
In Section 7, the author investigates the boundary case of overconvergent modular forms of slope one less than the weight. He shows that non-classical forms with this property exist but that any eigenform for the full Hecke algebra of weight \(k>1\) is classical if it does not equal \(\theta^{k-1}G\) where \(G\) is an overconvergent modular form of weight \(2-k\).
In Section 8, he proves a generalization of Theorem 6.1 (Theorem 8.1) which relates forms of level \(\Gamma_1(Np)\) to what we call overconvergent forms of level \(\Gamma_1 (Np)\) and in Section 9, he interprets these latter as certain Serre \(p\)-adic modular forms with non-integral weight. The central idea in this paper is expressed in Theorem 5.4 which relates overconvergent modular forms to the de Rham cohomology of a coherent sheaf with connection on an algebraic curve. More precisely, the author shows that there is a map \(\theta^{k+1}\) for non-negative \(k\) from modular forms of weight \(-k\) to modular forms of weight \(k+2\) which on \(q\)-expansions is \((qd/dq)^{k+1}\). When \(N>4\), the \(k\)-th symmetric power of the first relative de Rham cohomology of the universal elliptic curve with a point of order \(N\) over the modular curve \(X_1(N)\) is naturally a sheaf with connection. Theorem 5.4 is the assertion that the cokernel of \(\theta^{k+1}\) is the first de Rham cohomology group of the restriction of this sheaf to the complement of the zeros of the modular form \(E_{p-1}\) on \(X_1(N)\).
The above result, Theorem 6.1, is intimately connected with the conjectures of Gonuvêa and Mazur on families of modular forms. Indeed, in [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)] the author used it to deduce qualitative versions of these conjectures.

MSC:

11F85 \(p\)-adic theory, local fields
14G20 Local ground fields in algebraic geometry
11F33 Congruences for modular and \(p\)-adic modular forms
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References:

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