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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 p , N be a positive integer such that (N,p)=1 and k an integer. Katz has defined the space M k (Γ 1 (N)) of overconvergent p-adic modular forms of level Γ 1 (N) and weight k over K and there is a natural map from weight k modular forms of level Γ 1 (Np) with trivial character at p to M k (Γ 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)= n0 a n q n then

UF(q)= n0 a pn q n ·

Then the author proves (Theorem 6.1), that if F is a generalized eigenvector for U with eigenvalue λ (i.e., in the kernel of (U-λ) n for some positive integer n) of weight k and λ has p-adic valuation strictly less than k-1, then F is a classical modular form. In this case the valuation of λ 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 θ 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 Γ 1 (Np) to what we call overconvergent forms of level Γ 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 θ 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 θ 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.

11F85p-adic theory, local fields
14G20Local ground fields
11F33Congruences for (p-adic) modular forms