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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 ${ℂ}_{p}$, $N$ be a positive integer such that $\left(N,p\right)=1$ and $k$ an integer. Katz has defined the space ${M}_{k}\left({{\Gamma }}_{1}\left(N\right)\right)$ of overconvergent $p$-adic modular forms of level ${{\Gamma }}_{1}\left(N\right)$ and weight $k$ over $K$ and there is a natural map from weight $k$ modular forms of level ${{\Gamma }}_{1}\left(Np\right)$ with trivial character at $p$ to ${M}_{k}\left({{\Gamma }}_{1}\left(N\right)\right)$. 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 $F\left(q\right)={\sum }_{n\ge 0}{a}_{n}{q}^{n}$ then

$UF\left(q\right)=\sum _{n\ge 0}{a}_{pn}{q}^{n}·$

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 ${\left(U-\lambda \right)}^{n}$ for some positive integer $n\right)$ 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 $\left(k-2\right)/2\right)$. 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 ${{\Gamma }}_{1}\left(Np\right)$ to what we call overconvergent forms of level ${{\Gamma }}_{1}\left(Np\right)$ 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 ${\left(qd/dq\right)}^{k+1}$. 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 ${X}_{1}\left(N\right)$ 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 ${X}_{1}\left(N\right)$.

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 11F33 Congruences for ($p$-adic) modular forms